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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 CataloginginPublication Data. Die Deutsche Bibliothek

CIPEinheitsaufnahme
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 3540671269 ISSN 09407677 (Lecture Notes in Physics. Monographs) ISBN 3540671269 SpringerVerlag 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 semiclassical 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 DuistermaatHeckman 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 semisimple Lie groups, quantization of spin systems, unitary integrations in matrix models, modular invariants of Riemarm surfaces, supersymmetric quantum field theories, twodimensional YangMills 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 BatalinFradkinVilkovisky and MathaiQuillen formalisms, are presented and used to discuss the general relationship between topological field theories and more conventional physical models. ical

Copenhagen, September
Acknowledgements:
1999
Richard J. Szabo
I would like to thank I.
Semenoff for advice and encouragement book and for
helpful discussions.
I
during
Kogan,
G.
Landi,
F. Lizzi and G.
various stages of the
writing
the various calculational and
grateful conceptual aspects of Section 7,
portant historical remarks
the manuscript, and to 0. Tirkkonen for
on
am
to L. Paniak for his
of this
participation
in
to E. Gozzi for im
illuminating
VIII
Preface
discussions. I would also like to thank D. Austin and R.
suggestions supported Canada.
on some
in part
of the
more
Douglas for
comments and
mathematical aspects of this book. This work
was
by the Natural Sciences and Engineering Research Council of
Contents
1.
Introduction
2.
..............................................
1.2 1.3
Outline
............
1
..........................
5
...............................................
Equivariant Cohomology Principle 2.1 Example: The Height Function of the Sphere A Brief Review of DeRham Cohomology 2.2 2.3 The Cartan Model of Equivariant Cohomology and the Localization
2.4
Fiber Bundles
2.5
The
2.6
The
and
3.
7
............................
11
...............
11
..................
13
.............
19
....................
24
....................
33
.............................
38
Equivariant Characteristic Classes Equivariant Localization Principle BerlineVergne Theorem
Fin iteDimensional Localization for 3.1 3.2 3.3
Theory Dynamical Systems Symplectic Geometry Equivariant Cohomology on Symplectic Manifolds StationaryPhase Approximation
...................................
43
...................................
44
..........
47
and the DuistermaatHeckman 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 DuistermaatHeckman 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 AtiyahSinger 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 DuistermaatHeckman
and the NiemiTirkkonen 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 NonHomogeneous 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 NonSymmetric 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 SemiClassical
................
233
..................................
234
Geometrical Characterizations of the
Loop Expansion
7.2
Conformal and Geodetic Localization
7.3
Corrections to the DuistermaatHeckman 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 TwoDimensional YangMills 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 BatalinRadkinVilkovisky Formalism Equivariant Euler Numbers, Thom Classes and the MathaiQuillen Formalism The MathaiQuillen Formalism for InfiniteDimensional 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 lefthand 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 © SpringerVerlag 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 nontrivial 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
(27r e' .7r/2)
el1E.
i
e
I
Ei,jAi(MYjAj
v1det M
k=1
(1.2) where M
=
[Mij]
is
a
nonsingular 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 finitedimensional integral this is the wellknown stationary phase (otherwise known as the saddlepoint or steepestdescent) approximation [64]. In the framework of path integration, it is usually referred to as the WentzelKramersBrillouin (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 semiclassical 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 semiclassical approximation is exact, one can think of the Gaussian integration formula (1.2) as a 'localization' of the complicated looking integral on the lefthand 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 semiclassical 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 piecewiselinear 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 nontrivial examples of classically integrable models (i.e. ones whose classical equations of motion are 'exactly solvable'), for example the sineGordon model, where the semiclassical 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 HamiltonJacobi 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 semiclassical 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. 4dimensional YangMills 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 socalled 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 bosonfermion 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 AlvarezGaum6 [5], and Friedan and Windey [48], to give supersymmetric field theory proofs of the AtiyahSinger 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 nonabelian 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 4manifolds. 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
wellunderstood
[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. AharonovBohm 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 semiclassical approximation. It would be nice to put semiclassically exact features of functional integrals, as well as the features which reduce them to integrals over finitedimensional 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 semiclassical "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 finitedimensional
integrals'.
to Gaussian ones and essentially repreThe transition between the functional and
finitedimensional
integrals can then be regarded as a rather drastic localoriginal infinitedimensional 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 3dimensional 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 DuistermaatHeckman theorem
[39], which established the exactness
of the semiclassical
approximation for finitedimensional oscillatory integrals (i.e. finitedimensional versions of (1.1)) over compact symplectic manifolds
in certain instances. The DuistermaatHeckman 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.
DuistermaatHeckman 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 DuistermaatHeckman
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 infinitedimensional
generalization of the DuistermaatHeckman 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 DuistermaatHeckman 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 wellknown
=
AtiyahSinger
index theorem
correctly.
The crucial idea
was
the interpretation of the fermion bilinear in the supersymmetric action as a loop space symplectic 2form. 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
infinitedimensional generalization of the DuistermaatHeckman 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 wellknown semiclassical properties of these systems. It was the beautiful paper by Stone [156] in 1989 that brought the DuistermaatHeckman 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
DuistermaatHeckman 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., KaeMoody groups and the Virasoro group. The common feature is always that the path integrals are given exactly by a semiclassical Nielsen and Rohrlich from
a more
localization formula that resembles the DuistermaatHeckman 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 ChernSimons gauge theory using the
DuistermaatHeckman
integration formula. Based
on
this interpretation, and
[57][59] of the hidden supersymmetry underclassical dynamical system, Blau, KeskiVakkuri and Niemi [30] lying any introduced a general supersymmetric (or equivariant cohomological) frame
the observation of Gozzi et al.
investigating DuistermaatHeckman (or WKB) localization formulas generic (nonsupersymmetric) 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 BecchiRouetStoraTyupin (or BRST for short) quantization of constrained systems [16]. BRSTcohomology 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 nontriviality of their path integrals lie in thelarge kinematical nonlinearity 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 nonlinear) 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 finitedimensional integral in analogy with the classical Gaussian integration formula (1.2), we shall first analyse the symmetries responsible for the localization of finitedimensional 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 infinitedimensional 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 loopexpansion 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 nonlocalizable 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 selfcontained 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 2sphere S2 of unit radius viewed in Euclidean 3space R as a sphere and centered at z a symmetrically about standing on end on the xyplane the zaxis. 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 3space 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 pform a and a qform 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
gradedcommutative algebra called the exterior algebra of M. In product of a pform a and a qform 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
kforms
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 kforms modulo exact kforms is called the kth 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 Ak1M. tial kforms where 2 differential forms
are
are
One important theorem in DeRham
cohomology is Poincar6's lemma. This starshaped 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 qdimensional 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
qsimplex. 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 qsimplex 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 ith one, and the sum on the righthand side is the formal algebraic sum of simplices (where a minus sign signifies a change of orientation). A singular
qsimplex
of the manifold M is defined to be
A formal
algebraic
a
continuous map
a :
Aq
M.
of
qsimplices with integer coefficients is called a qchain, and the collection of all qchains 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 qchain 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)
+
Cql(M)
(2.30)
18
Equivariant Cohomology
2.
and the Localization
Principle
of abelian groups. The collection of abelian groups fCq(.A4)lqcz+ and nilpohomomorphisms 0 form the singular chain complex C,, (M) of the man
tent
ifold M.
Nilpotency
of the
boundary
(2.30)
that every
qchain in the qboundaries of M, lies as well in the kernel of Olc,, whose elements are called the qcycles of M. The abelian group defined as the quotient of the group of all qcycles modulo the group of all qboundaries is called the qth (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 qcycle (intuitively a closed qdimensional curve or surface) encloses a q + 1dimensional chain and M has no 'qholes'. For instance, if M is simplyconnected (i.e. every loop in M can be contracted 0. A starshaped region, such as a simplex, is to a point) then H, (M; Z) simplyconnected. Given the abelian groups (2.31), we can form their duals using the uniinvariants and
bicontinuous
=
=
versal coefficient theorem
Hq (M;
Z)
_
Homz (Hq(M; Z), Z)
E)
Extz(HqI(M; Z), Z)
(2.32)
which is called the
qth 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 qdimensional boundary ac of c. The R defines a natural duality f w on Hq (M; R) 0 Hq (M; R)
which relates the in M to
as an
vector space
ac
C
c
a
of DeRham's theorem is Stokes'
wEAqM
(2.33)
+
C
2.3 The Cartan Model of
Equivariant Cohomology
pairing between Hq (M; R) and Hq (M; R) and isomorphism. In particular, (234) generalizes to theorem,
I
dw
w
m
which relates the
(n
closed

w
c
19
is the basis of the DeRham the
global
version of Stokes'
A''M
(2.35)
am
integral
of
exact form
an
over
M to
an
integral
over
the
I)dimensional boundary
ifold is defined
by
aM of M. Here integration over a manpartitioning the manifold up into open sets homeomorphic
n
integrating a top form (i.e. a differential form of highest degree n on M) locally over Rn as usual', and then summing up all of these contributions. In this way, we see how the DeRham cohomology of a manifold measures its topological (or global) features in an analytic way suited for the differential calculus of C' manifolds. We refer to [98] and [32] for a more complete and leisurely introduction to this subject. to R
,
2.3 The Cartan Model of We shall
now
where there is whose group
the constructions of the last Section to the
generalize a
Equivariant Cohomology
Lie group
(i.e.
multiplication
a
continuous group with
is also
smooth) acting
case
smooth structure
the space. Then the (i.e. structures that are
on
construction of topological invariants for these spaces
the
a
homeomorphic spaces) will be the foundation for the derivation general integration formulas in the subsequent Chapters. Many situations in theoretical physics involve not only a differentiable manifold M, but also the action of some Lie group G acting on M, which we denote symbolically by GxM>M same
for
of
(2.36)
(g, X)

g

X
By a group action we mean that g x x, Vx G M if g is the identity element G, and the group action represents the multiplication law of the group, i.e. 91'(92'X) (9192) X7 V91 92 C G. We shall throughout assume that G is connected and that its action on M is smooth, i.e. for fixed g E G, the function x + g x is a diffeomorphism of M. Usually G is taken to be the symmetry group of the given physical problem. The common (infinitedimensional) 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
2manifold)
p4orm. with
and G is the semidirect
p < dim M is
product
of
always understood here
to
20
the
2.
Equivariant Cohomology
and the Localization
Principle
Weyl
and diffeomorphism groups of that 2surface. 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 Gequivariant 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 Gaction 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 Gequivariant cohomology is defined the as simply cohomology of the coset space MIG, of M. Given the Gaction
=

=


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 fixedpoint free coordinate neighbourhoods (because then the isotropy subgroup Gx of that fixed point x is nontrivial), 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 YangMills theory in 4dimensions 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 Gactions on them is equivariant with respect to the group action if
f(gx) =g.f(x)
VXEM1
,
VgEG
(2.38)
2.3 The Cartan Model of
Equivariant Cohomology
21
Wewant to extend this notion of equivariance to differential forms. Consider the symmetric polynomial functions from the Lie algebra g of G = exp(g) into the exterior algebra AM of the manifold M. These maps form the algebra
S(g*) (&AM,
where
S(g*)
is called the
symmetric algebra over the dual vector
space g* of g and it corresponds to the algebra of polynomial functions g. The action of g E G on an element a E S(g*) 0 AM is given by
(g a) (X) 
where X
Here
g
=
(a (g'Xg))

have used the natural
on
(2.39)
coadjoint
action of G
on g* by the tensor transformation law (2.12) with x'(x) g x. From this it follows immediately that the equivariance condition (2.38) is satisfied for the polynomial maps a: g > AM in the Ginvariant subalgebra
E g.
we
and the induced Gaction
AM from that
on
=
A,M where the
(2.40)
are
Elements of G
are
g are
as
dictated
0
AM)'
(infinitesimal)
(2.40) Ginvariant 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 Gaction
on
M
can
be
(ad
Xa )be
=
represented locally
if abc. as
the continuous
flow 9t

X
=
X(t)
t E
R+
(2.43)
where gt is a path in G starting at the identity gt=o. The induced action differential forms is defined by pullback, i.e. as
(gt a)(x) 
For
==
a(x(t))
on
(2.44)
example, we can represent the group action on C' functions by diffeomorphisms on M which are connected to the identity, i.e.
2.
22
Equivaxiant Cohomology and the Localization Principle
(gt f ) (x)
=

The action where
V(x)
element. It
f (x (t))
etv ('(t)) f (x)
=
f
,
AO M
E
(2.45)
(2.45) represents the flow of the group on C' functions on M, VA(x) ax" a is a vector field on M representing a Lie algebra is related to the flows (2.43) on the manifold by =
 'W which defines
a
set of
=
VIWO)
in M which
curves
(2.46)
will refer to
we
as
the
integral
curves
of the group action. If VI is the vector field representing the generator X1 of g, then the Lie algebra (2.42) is represented on C' functions by
[Va' Vb] (h)
fabcVc(h)
=
Vh
E
AOM
(2.47)
represented by the ordinary commutator bracket. This derepresentation of G by vector fields in the tangent bundle TM. In this setting, the group G is represented as a subgroup of the (infinitedimensional) 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
kforms
Wcellvl
AkIM
(2.49) multiplication, with respect
(215) 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 Gaction the Lie derivative
on
along
the
higherdegree differential forms
is
generated by
V
Lv: A kM
+
AkM
(2.51)
where
Lv
=
generates the induced action of G
Lva(x(O)) This
can
be verifed
using (2.12)
and
by
direct
(2.46),
and
(2.52)
div + ivd
on
AM, d
=
i.e.
wt a(x(t))
lt=O
computation from expanding (2.44) about by noting that
(2.53) t
=
0
2.3 The Cartan Model of
[,CV. "CVb ] (a) Thus the Lie derivative in
=
Equivariant Cohomology
f,b,,CV. (a)
general defines
Va
,
a
E
23
AM
(2.54)
representation of G
on
AM.
The local components of LvT for a general (k, t) tensor field T are found by substituting into the tensor transformation law (2.12) the infinitesimal
coordinate field W
=
change x'A(x)
39 axl,
WA (x)
(LVW? Furthermore, on
=
xP(t)
=
xg +
tVl'(x).
For
example,
on a
vector
have
we
=
W'09,V"
V,09,WA

the Lie derivative Lv is
an
=
[WI VIII
(2.55)
ungraded derivation
and its action
contractions is
[iVa,,CVbl (a)
=
fabc iv. (a)
(2.56)
We are now ready to define the Cartan model for the Gequivariant cohomology of M [21, 33, 99]. We assign a Zgrading2 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= (10dOaOiV. )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 CartanWeil 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 Ginvariant algebras lAkGM}kEz+ and nilpotent A
Zgrading is usually refered to as a 'ghost number' in the physics literature. equivalence between the 2 notions will become clearer when we deal with path integrals in Chapter 4 see Appendices A and B for this algebraic correspondence. The

24
2.
Equivariant Cohomology and the Localization Principle
derivations D 9 thereon defines the
Gequivariant complex A*(M) G
of the
manifold M.
Thus, just
in the last
as
Section,
we can
proceed
to define the
of the operator Dg. The space of equivariantly closed modulo the space of equivariantly exact forms, i.e. a
Gequivariant cohomology TTk "G(M)
With this
definition,
free Gaction space
MIG,
known
(2.60)
as
=
in
(2.37).
Dg,8,
Dga
=
0,
is called the
DgIA M/im DglAkim
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
fixedpoint
cohomology
of the quotient of equivariant cohomology is that the definition of
Dg
in
gaugecovariant 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 Ginvariant 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 topform component a(') E AIM a
=
=
ordinary DeRham sense. This follows because the iv part of Dg lowers the formdegree by I so there is no way to produce a toPform by acting with iv. Finally, in what follows we shall have occasion to also consider of AGM to include arbitrary Ginvariant smooth the C' extension AIM G functions from g to AM. In this extension we lose the Zgrading described above, but we are left with a Z2grading corresponding to the differential form being of even or odd degree [21] (Z2 Z/2Z is the cyclic group of order is exact in the
=
2).
2.4 Fiber Bundles
and
Equivariant Characteristic Classes
The 'bundles'
examples
we
introduced in Section 2.2
(tangent, cotangent, etc.)
are
all
general geometric entity known as a fiber bundle. The geometry and topology of fiber bundles will play an important role in the development of equivariant localization theory, and in this Section we briefly review the essential features that we shall need (see [22, 41, 341 for more detailed discussions). A fiber bundle consists of a quadruple (E, M, F,7), where E is a topological space called the total space of the fiber bundle, M is a topological space called the base space of the fiber bundle (usually we take M to be a manifold), F is a topological space called the fiber, and 1 7r : E > M is a suriective continuous map with 7r (x) F, Vx E M, which is called the projection of the fiber bundle. A fiber bundle is also defined so of
a more
=
2.4 Fiber Bundles and
that
locally it is trivial, neighbourhood U C M
Equivariant Characteristic Classes
25
locally the bundle is a product U x F of an open fibers, and 7r : U x F + U is the 3. the In first coordinate onto the case of the tangent bundle, for projection the fibers T R' F and the projection map is defined _ are .,R' instance, on TM + M by 7r: T ,M + x. In fact, in this case the fibration spaces are vector spaces, so that the tangent bundle is an example of a vector bundle. If the fiber of a bundle is a Lie group G, then the fiber bundle is called a principal fiber bundle with structure group G. It has a right, smooth and i.e.
of the base and the
=
free action of G
the total space E and the base M which gives
on
representation of the group in the fibers. This action also embeds the group'G inside E. The vector and tensor fields introduced in Section 2.2 should be defined
cisely s
:
M
+
Although
as
'sections' of the associated
E which take
shall be
a
point
bit abusive in
a
local
copy of
more
pre
i.e. smooth maps
M into the fiber
E
X
bundles,
a
7r'(x)
over
discussion
by considering M, for simplicity and ease of notation, it should be "kept in mind that it is only locally where these objects admit such a functional interpretation. Thus, for instance, the tangent bundle is TM J(x, V) : x E M, V E TxMJ and locally a section of TM can be x.
these
as
we
a
genuine functions
our
on
=
(x, V4 (x) .0. ). (i.e. bases) on the tangent bundle TM form a principal GL(n,R)bundle over M, called the frame bundle, whose points are (x; (ei,..., en)) where x E M and (ei,..., en) is a linear basis for TxM. If M has a.metric (i.e. a globally defined inner product on each tangent space TxM), then we can restrict the basis to an orthonormal basis and obtain a principal 0 (n N, N)bundle, where (n N, N) is the signature of the metric. written
as
X
The set of frames


If M is furthermore
orientable, then we can further restrict to an oriented ordefined basis, by the equivalence classes with respect to the equivthonoi'tmal alence relation e = M f where det M > 0, and get a principal SO (n N, N)bundle. When M is a spacetime 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
Gbundle 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 Gaction
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(g1) f (p). Notice that for a vector bundle E M, the bundle
differential
f (p g) 
=
forms)
on
+
+

of differential forms
on
M with values in E is defined
Ak (M, E)
=
as
A kM 0 E
(2.63)
products and direct (Whitney) sums of bundles are defined locally by corresponding algebraic combination of their fiber spaces. Intuitively, a fiber bundle 'pins' some geometrical or topological object where tensor
the
over
each point of
bundle).
manifold M
a
vector space W
bundle is
=
(e.g.
a
vector space in the
case
of
a
vector
R', then the tangent bundle TR' associates the R' to each point of R'. In fact in this case, the tangent
For instance, if M
=
globally given by TR'
R'
=
x
W, the product
of its base and
fibers. We then say that the bundle is trivial, in that the erecting of points into vector spaces is done without any 'twistings' of the fibers. However, a
general
vector bundle is
only locally
trivial and
the fibers
globally
twist
can
very complicated fashion. One way to characterize the nontriviality of fiber bundles is through special cohomology classes of the base manifold M
in
a
[104].
called characteristic classes
signifies
sense
As
the
nontriviality
A nontrivial 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 nontriviality 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 Gequivariant bundle if there are Gactions 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
Gequivariant
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 1form) 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 1form in Al (M, E). If V is a tangent vector on M, its action on a section s is defined in local coordinates by
2.4 Fiber Bundles and
(VS)"M where
(.P,,)Pdx"
the
path
is
1form
a
E).
space indices in
on
If x(t) is
a
=
Equivariant Characteristic Classes
V109AS"
+
(FOPSO)
M with values in
path
(2.66)
End(E) (a,,3
M, then: (t)
in
27
is
a
are
tangent
the vector
vector
along
and the equation
Ts)(: M) determines
parallel transport along
the
(2.67)
0
=
path allowing
us
to connect different
fibers of the bundle. The first order differential equation (2.67) can admit topologically nontrivial solutions if either the space M is multiplyconnected
(Hi (M; Z) 54 0), or if the connection F has nontrivial curvature F 54 0 (see below). The latter condition characterizes the nontriviality 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 Gequivariance mean, respectively, =
=
Cvl'
X on
P
adjoint finite, total adjoint
r,.g
on
briefly
G and G
look at
then 
pt
=
a
=_
E g
[x, ri and
action of the group
Ad(g')r
=
(2.68)
ad(X)
on
r
denotes the
be
can
exponenti
as
girg
(2.69)
examples. If G is a matrix group (e.g. SU(N), regard matrix elements of g E G as functions principal Gbundle. Then the unique solution to
some
as
a
(2.68)
for
a
Lie group is called the CartanMaurer
1form 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 Gequivariant
we

where V is the vector field
Let
principal Gbundle,
a
under the action of g E G). In this case, Vp g so that the 1form with values the Lie in a globallydefined algebra g, i.e.
iVI'
infinitesimal
M is
>
into horizontal and vertical components be
general Gbundle
P
=
") M, in
gldg 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
gldg + g'A,,gdx4 dxA is
a
Lie
refered to in this context
as
algebra a
(2.71) valued 1form
gauge connection
on
or
M,
gauge
28
Equivariant Cohomology and the Localization Principle
2.
field. The transformation laws
by
A
>
local
patch boundaries on M, labelled (the automorphisms of the bundle), act on
across
Gvalued transition matrix g gauge connections via pullback, a
Ag
_=
g'Ag + gldg
(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
CivitaChristoffel 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 2form
=_
(2.73)
V2
is
=
a
and
so
principal Gbundle
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,
g1Fjv(x)gdx1' A dx'
F
+
it
can
be
regarded
=
(g1X'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 Gequivariant bundle, Ginvariant,
is
[V,,Cvl Mimicking the equivariant
F]
A
we assume
0
(2.77)
exterior derivative
(2.58), we define the equivariant
=
covariant derivative
Vg
=
1 (& V

Oa
as an
differential forms
M with values in E. In
U C
M,
this
algebra
operator
looks like
on
the
a
(2.78)
iV.
algebra Ag (M, E)
which is considered on
(&
of
equivariant U x W,
local trivialization E
=
2.4 Fiber Bundles and
AG(U, E)
Equivariant Characteristic Classes
(S(g*)
=
0 AU o
Recalling the CartanWeil 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 Gaction with respect to the connection V. The moment map /,t is a Gequivariant extension of the ordinary curvature
(2.73)
2form
equivariant
from
one
a
When evaluated V E TM 0
W,
covariantlyclosed 2form,
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 Gaction 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 (281) we see that it obeys the important property can
*
Vttv so
that
a
nontrivial moment map
the curvature of the connection V
=
ivF
produces
(2.86)
a nonzero
(c.f. (274)).
vertical component of we shall encounter
Later on,
2 important instances of equivariant bundles on M, one associated with a Riemannian structure, and the other with a symplectic structure. In the latter case the moment map is associated with the Hamiltonian of a dynamical
system. Now
we
class. First, [104]. Given
are we
a
ready
to define the notion of
equivariant characteristic
Lie group H with Lie algebra h, we say that a real or complexan invariant polynomial on h if it is invariant under the
valued function P is natural
an
recall how 'to construct conventional characteristic classes
adjoint
action of H
on
h,
30
2.
Equivariant Cohomology and the Localization Principle
P(h'Yh)
=
P(Y)
Vh E
H,VY
E
(2.87)
h
An invariant
polynomial P can be used to define characteristic classes on principal fiber bundles with structure group H. If we consider the polynomial P in such a setting as a function on hvalued 2forms on M, then the Hinvariance (2.87) of P implies that
dP(a) where
is the
r
degree
curvature 2form
is
locally
an
a
=
hvalued
=
rP(Va)
of P. In F
=
E
A 2M 0 h
(2.88)
particular, taking the argument a to be the the principal Hbundle E ") M (which
V2
2form),
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 CartanKilling linear form of the just P(a) Lie algebra h (usually the ordinary operator trace). Consider a continuous V2t oneparameter 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] n1
Ft
dt
d
=
=
d tr
n1
d
dt
n1
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 2nform 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 Hbundle. 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 Ginvariant polynomial P, (2.89) generalizes to
P(F)
E
Occasionally, forms
choice of connection. In
AM is called
as
for
ease
a
characteristic class of the
of notation,
ordinary multiplication.
we
shall denote exterior products of differential
For
instance,
we
define
ce
An
=
Cen.
2.4 Fiber Bundles and
DgP(Fg) and
Equivariant Characteristic Classes
rP(VgFg)

=
31
(2.92)
0
the
resulting equivariant characteristic classes P(Fg) of the given Gequivariant 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(Itv)
=
P(p(X))
(294)
where p is the representation of G defined by the Gaction on the fibers W. There are 4 equivariant characteristic classes that commonly appear in the localization formalism for be understood and
as
extensively discussed polynomial tr ec, and is used one
topological field theories, all of which are to completion A M. These can all be found
elements of the
are
in which the fibers
are
[211.
in
for
The first
one
is related to the invariant
Gequivariant complex vector bundles (i.e. over the complex numbers C). It is
vector spaces
called the Gequivariant 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 Agenus
a
G
vector bundle
Ag(F)
det sinh(!Fg)] 2
=
F9
(2.96)
2
where the inverse of
always
an
inhomogeneous polynomial
of differential forms is
to be understood in terms of the power series
(1
+
X)1
=
1:(_I)kXk
(297)
k=O
On
a
is the
complex fiber bundle, the complex equivariant Todd class
tdg(F) 5
A n6ntrivial vector bundle with
a
can
nontrivial curvature F
applications
to
topological
:34
=
det
always
[
version of the
eFg

11
be considered
0. This
point of
gauge theories.
equivariant
Agenus
(2.98) as a
view is
trivial
one
quite useful
endowed
in certain
32
2.
Equivariant Cohomology
and the Localization
Principle
When G is the trivial group, these all reduce to the conventional characteristic classes [104] defined by replacing Fg + F in the above. Just as for the
ordinary Agenus 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)
=
SalamMathiews
[Mij]
metric matrix M
td
class,
Eg (F) where the Pfaffian
=
9
is defined
determinant)
of
a
2N
x
2N
antisym
as
Mi2N1i2N
Mili2
E
(2.100)
N
1:
_N N!
sgn(P)
(2.101)
11 MP(2k1),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 skewdiagonal
eigenvalues
(2.101),
in
a
skewdiagonalization
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 2form 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 GaussBonnet theo
arbitrary
vector bundle
2.5 The
E(F)
with Euler class
of
Equivariant Localization Principle
curvature 2form F E
a
A'(M, E)
33
is called the
GaussBonnetChern 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 Hbundle over a 2kdimensional manifold M, the integral over M of the kth term in the expansion of the conventional version of the Chern +
=
(2.95) (which
class
defines the kth 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 kth Chern number of M (or, precisely, of the complex vector bundle (E, M, W, 7r)) The Chern number is always an integer for closed orientable manifolds. Thus the equivariant; characteristic classes defined above lead to interesting equivariant generalizations of some classical topological invariants. In the next Chapter we will see that their topological invariance in both the ordinary DeRham and the equivariant cases are a consequence of the topological invariance of the integrations there. We shall see later on that they appear in most interesting ways within the formalism of localization formulas and topological field theory functional integration. more
.
2.5 The We
now
Equivariant Localization Principle
discuss
a
very
interesting property of equivariant cohomology which
is the fundamental feature of all localization theorems. It also introduces
the fundamental
constraint that will be
'geometric
in what follows. In most of
following
situation. Let M be
ary and let V be
circle group G
0
E
S(u(l)*), U(1),
bra of
a
=
a
vector field
U(1)

which is
will not be
a
applications
our
S'
we
one
of the issues of focus
will be concerned with the
compact orientable manifold without boundM corresponding to some action of the
over
M. In this
on
linear functional
case on
the role of the
multiplier algefollows. Indeed,
the 1dimensional Lie
important for the discussion that
regard 0 as just some external parameter in'this case and 'localize' 1. As shown in [9] (see also Appendix B), algebraically by setting 0 the operations of evaluating 0 on Lie algebra elements and the formation of equivariant cohomology commute for abelian group actions, so that all results below will coincide independently of the interpretation of 0. In particular, for we can
=
a
free
U(1)action
on
(S(u(l)*) so
that in this
case
M
0
we
have
AM)U(1)
the
=
S(u(l)*)
(9
A(MIU(1))
(2.106)
multipliers 0 play no cohomological role and the restricts to the cohomology of the quotient space
equivariant cohomology just
34
Equivariant Cohomology and
2.
MIU(1).
The
the Localization
corresponding equivariant
Principle
exterior derivative is
now
denoted
as
D,,,(:L) and it is
now
considered
as an
AvM It
was
Atiyah
=
operator
fa
=
[9]
and Bott
Dv
=
d + iv
on
the
algebra
AM:,Cva
E
(2.107)
01
=
and Berline and
(2.108)
Vergne [19, 20] who first point locus
noticed that equivariant cohomology is determined by the fixed of the Gaction. 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 fixedpoint 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 topform component of a is exact. Since integration over M picks up the topform 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 globallydefined U(1)invariant Riemannian structure on it, which means that it admits a globallydefined
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,Xq,V1 + g ,,Xa,,VA + Oaxgl"
=
0
(2.112)
2.5 The
Alternatively,
Equivariant Localization Principle
this Lie derivative constraint
g"XV1.V1'
gA,\VVV,1
+
(2.65)
where V is the covariant derivative
be written
can
35
as
(2.113)
0
=
constructed from the LeviCivita
Christoffel connection
_VA
/w
associated with g
on
9" (.9Ag" + '9vg"
2
(2.114)
'9PgA')

the tangent bundle TM. Here gAV is the matrix inverse on the vector field V in the usual
of gA, and the covariant derivative acts way
as
V A VV
19AVV
=
VM VA
+ PA
(2.115)
I
plus sign for (0, k)tensors and a minus sign for (k, 0)tensors in front in (2.115). Notice that by construction the Levi CivitaChristoffel r,\ and it is compatible with the the metric connection is torsionfree, 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 (Ginvariant) 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 globallydefined C' functions on M,
whose components gAv (x) g,, (x) is the crux of all finite and infinitedimensional 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 1forms, 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 1form
#
=
g (V,
)
=
gA, (x) V' (x) dxA
(2.117)
36
2.
Equivariant Cohomology and the Localization Principle
Nondegeneracy, det g(x) :7 0, VX an isomorphism between The Iform,6 satisfies
M, of the
E
defines
D 2,3 V
=
fVa
=
LVV Killing equivariant differential 1form. Furthermore, since
=
0 and V is
globallydefined Kv
(2.118)
=
Kv
+
we
that 3 is
means
an
have
Ov
(2.119)
CIfunction
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 2form 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 0form DV,3 is invertible on M MV. an inhomogeneous differential form
and hence
nonzero
understand here the inverse of
nonzero
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 topform 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 > Ank(pt)
the
on
Equivariant Localization Principle
on
map
=
Hn (.A4;
R)
for
+
HO (pt; R)
=
R. The
same
is true for
equivariant
integration. Since, general M, integration of a differential form picks up the topform component which for an equivariantly exact form is exact, for any equivariantlyclosed differential form a we can again invoke Gaction
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 nonabelian integration
Later on,
case as
we
over
in the
a
shall also consider the dual Lie
well.
Given that the class defined
by
a,
fm a depends only on the equivariant cohomology
integral we can
choose
a
particular representative of the cohomolTaking the equivariant differential integral
ogy class making the localization manifest. form,3 defined in (2.117), we consider the
Z(S)
f
=
a
e Dv,6
(2.128)
M
viewed of
S E
as a
function of
R+ and that
its
s
E
s
+
R+.
We
0 and
s
*
oo
that
(2.128)
is
a
regular
limits exist. Its
s

assume
function
0 limit is the
interest, fM a, while from the identities (2.119) and (2.120) we see integrand of (2.128) is an increasingly sharply Gaussian peaked form around Mv C M as s + oo. The crucial point here is that the equivariant differential form which is the integrand of (2.128) is equivariantly cohomologous to a for all s E R+. This can be seen by applying Stokes' theorem to
integral
that the
get
of
38
Equivariant Cohomology and the Localization Principle
2.
d
a(Dv,3) esDv)3
Z(S) M
f f Dv(ao esDvO) +,3Dv (a esDv,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 nontrivial 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
BerlineVergne
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
BerlineVergne Theorem
39
method that will generalize to functional integrals. For now, we assume the fixedpoint 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 nonzero dimension later on. If we assume that M is compact, then MV is a finite set of points. We wish to evaluate explicitly the righthand 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 kth 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 kform component
with is
+
sum
Clfunction
on
of
a
a(n)
and
a(k)
a(0) (x)
E
AkM
its 0form
M. The kform 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 2ndimensional
now
(2.133)
k>O
of view differential forms
exterior bundle which is
The

a(x,,q)
on
the
0 AM
supermanifold M
(x,,q).
integration of a differential form is now defined by introducing the rules for integrating Grassmann variables [17],
f d77A
IR
A
=
f dql'
1
1
=
(2.134)
0
71"'s are nilpotent, any function of them is a polynomial in 77" consequently the rules (2.134) unambiguously define the integral of any function of the anticommuting variables 77". For instance, it is easily verified Since the and
that with this definition of We shall 7
If
we
introduce
formula
here that
assume
but it will allow
integration is
even.
have 7
This restriction is
by
no means
necessary
of the arguments in this Section. second independent set 1 1' I of Grassmann variables, then the
us
a
n
we
to shorten
(2.135) generlizes
to
some
arbitrary (not necessarily even) dimensions
n as
40
2.
Equivariant Cohomology
and the Localization
dnq el'7"M111"7' d?7' dq'1
_
Principle
Pfaff M
(2.135)
d771. Note that under a local change of basis qA algebra the antisymmetry property (2.131) and the Berezin rules (2.134) imply that f dni7 det A f dnq. It follows from this that the Berezin integral in (2.135) is invariant under similarity transformations. (2.135) is the fermionic analog of the Gaussian integration formula (1.2). The differentiation of Grassmann variables, for which the integration in (2.134) is the antiderivative thereof, is defined by the anticommutator where
dn,
=
...
A Vlq' of the Grassmann
+
19
aqA With these
,77V
I
=
R
(2.136)
A
+
definitions, the integration by parts formula
f dq"
d
d
d7711
f(ntt)
=
0
f dqA f(,qli) &71A gqji). definitions, we can now alternatively write the integral of any differential form over M as an integral over the cotangent bundle M OA1M. Thus given the localization formula (2.130) with the Iformfi 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
...
larges
(&A' M
M

2
Wn"n')
is
coordinateindependent because dnq transform inversely to each
A dXn and
limit of
(QV) w,
(2.137),
we
use
the deltafunction
representations 5 (V)
lim.
=
s
Jim 500
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),,,
Vldetg 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
efI"M`7'
=
detM
as
2.6 The
f
a
f
(,r)n/2
=
M
dnx
dnq a(x, 71)
Pfaff f2v (
M(DA1M
where
VFdetg(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).
BerlineVergne
(2141)
kills off all kform components of the form a except its Clfunction 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 VFdetg(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 deltafunctions EPEM, J(x p) localizing onto the zero locus MV. Substituting in the idendV(p), the tity (2.140) and noting that at a point p E Mv we have VV(p) expression (2.142) reduces to
where the factor formation
x

=
1
a=
PEMv
M
where
emphasize the
we
a
(27r)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 (nondegenerate 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 ndimensional our
first
discrete set of points in M and it is valid for any differential form a on a manifold with a globallydefined
a sum over a
equivariantlyclosed
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 finitedimensional integrals. We close this Chapter by noting the appearence of the operator in the denominator of the expression (2.143). For each p E MV, it is readily seen circle action
(and
generator is
a
Riemannian metric for which the associated
that the operator dV(p) appearing in the argument of the Pfaffian in (2.143) is just the in vertible linear transformation Lv(p) induced by the Lie derivative the tangent spaces TpM, i.e. by the induced infinitesimal group the tangent bundle (see (2.55)). Explicitly, this operator is defined vector fields W W"(x) .9x" Jx=p E TpM by
acting
on
action
on
on
2.
=
Lv(p)W Note however that
right
on
dV(p)
=
aV"(p)W'(p)
is not covariant in
the tangent space
TpM
a ax"
(2.144) X=P
general and so this is only true general on the entire tangent
and not in
42
2.
Equivariant Cohomology and the Localization Principle
bundle TM. A linear transformation
on the whole of TM can only be induced by introducing a (metric or nonmetric) 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. FiniteDimensional 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
finitedimensional
rigorous mathematical theorems concerning their behaviour
and therefore
be formulated. In the infinitedimensional cases, although the techniques used will be standard methods of supersymmetry and topological field theory, can
a
lot of rigor is lost due the illdefinedness of infinitedimensional manifolds integrals. A lot can therefore be learned by looking closely at
and functional some
finitedimensional
cases.
We shall be interested in certain
resenting
the
FourierLaplace
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 largeT 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 (infinitedimensional) symplectic manifold
(or phase space) [22]
and
we
shall therefore restrict
the remainder of this Book to the localization over
symplectic manifolds.
R. J. Szabo: LNPm 63, pp. 43  75, 2000 © SpringerVerlag Berlin Heidelberg 2000
theory
for
our
attention for
oscillatory integrals
44
1 FiniteDimensional 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 2ndimensional 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 nontrivial 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 coordinatefree 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
globallydefined nondegenerate closed 1 W
symplectic
form of M.
By closed dw
or
even
dimension
2form.
w,,, (x) dxA A dxv
=
2
called the
classical mechanics.
differentiable manifold A4 of
=
(3.1)
we mean as
usual that
0
(3.2)
in local coordinates
9mwv,\ Thus
w
defines
a
DeRham
+
19VWA1Z
+
a,\Wvv
cohomology class
=
0
in H 2(M;
(3.3) R). By nondegenerate
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
1form
a
0
0,,(x)dx"
(3.5)
dO
(3.6)
=,900, ,9,00
(3.7)
=
such that
or
in local coordinates
wo, The
locallydefined
1form 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 2form 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 2form determines a bilinear function I., J, AOM called the Poisson bracket. It is defined by
If, gl,, or
w
=
1
(df, dg)
f,g
E
AOM
:
AOM
(3.11)
in local coordinates
If, gl",
=
W", (490f (X),9,g(x)
(3.12)
where wA' is the matrix inverse of wjAv. Note that the local coordinate functions themselves have Poisson bracket I
The Poisson bracket is
IXA' X'},,,
=
antisymmetric,
W/" (X)
(3.13)
46
3. FiniteDimensional Localization
If, 0", it
obeys
Theory
for
Dynamical Systems
1 g' f 1"'
(3.14)
the Leibniz property
I f, ghl,, and it satisfies the Jacobi
gf f, hj,,
=
+
hf f, gl,,
+
I h, I f, gj,, j,,,
(3.15)
identity
I f, Ig, hl,, J,,
+
I g, I h, f j,, 1,,
=
(3.16)
0
This latter property follows from the fact (33) 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 2form 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 realvalued functions H on M (i.e. elements of AOM) will
where both ables
canonical momentum and
and
M.
be called classical observables. Exterior
products
mine nontrivial closed 2kforms
(i.e.
[Wk]
E
on
M
H2k (M; R)). In particular, the 2nform
of
nonzero
W
with itself deter
cohomology
classes
3.2
Equivariant Cohomology
dlZL 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(nl)/2,n /n!
3.2
=
dpl
A
...
Equivariant Cohornology
In this Section
dPn
A
on
A
dql
A



A
dqn
(3.23)
Symplectic Manifolds
specialize the discussion of Chapter 2 to the case a symplectic manifold of dimension 2n. Consider the action of some connected Lie group G on M generated by the vector fields VI with the commutator algebra (2.47). We assume that the action of G on M is symplectic so that it preserves the symplectic structure, we
shall
where the differentiable manifold M is
'CVW 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 1form the
symplectic
0. If 0 also satisfies
Lv,O
=
(3.26)
0
Ginvariant, and aca Gequivariant bundle. By definition (see Section 2.4) the structure group of this symplectic line bundle acts by canonical transformations. As such, w represents the first Chern class of this U(I)bundle, and, if M is closed, it defines an integer 2 cohomology class in H (M; Z) (as the Chern numbers generated by w are then the associated covariant derivative V
cording
then
to the
general
=
d + 0 is
discussion of Section 2.4 this defines
integers).
The associated moment map H : M + g* evaluated on a Lie algebra element X E g with associated vector field V is called the Hamiltonian
corresponding
to
V, Hv
From
(3.6)
and
(3.26)
=
Cv

[iv, V]
equivalently
map since
w
ivO
=
W'Op
this follows from the
=
ivw
(3.28)
general property (2.86) of the moment coordinates, this
is the curvature of the connection 0. In local
last equation reads
(3.27)
it then follows that
dHv or
=
48
3. FiniteDimensional Localization
,01,Hv(x) In
particular,
Theory
for
Dynamical Systems
V'(x)w,,,(x)
==
(3.29)
the components H' of the moment map H
Oa
=
(9 Ha
(3.30)
iv.w
(3.31)
satisfy dHa
=
Comparing symplecticity (3.25) on the group action, we see that this is equivalent to the statement that the closed 1forms 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 globallydefined 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 timeevolution) of a Hamiltonian vector field V as in (3.29) define the Hamilton equations of shall call the Gaction
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 timeevolution) 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 (317)
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, coordinateindependent 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 2form,
(,,+Oa
=
o H
a
(3.35)
have
(Dgwg) (X)
=
(d

iv) (w
+
Hv)
=
0
(3.36)
3.2
which is
equivalent
Equivariant Cohomology
(3.28)
to the definition
Symplectic Manifolds
on
49
of the Hamiltonian vector field
V. In fact, the extension (3.35) is the unique equivariant extension of the symplectic 2form w [129], i.e. the unique extension of W from a closed 2form
equivariantlyclosed one. Thus, we see that finding an equivariantlyequivalent to finding a moment map for the Gaction. 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 2form 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 =,CvH
=
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)
=
2cocycle
+
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 2cocycle Cab in (3.39) is in fact related to the noninvariance of the symplectic potential under G (c.f. eq. (3.26)).
possible
From the
symplecticity (3.24) ,Cv0
of the group action and
=
(ivd + diVa)O=
locally in a neighbourhood JV in M locallydefined linear functions consistency condition
the
=
it follows that
(3.41)
a
w
=
dO and Va
9(Xa)
=
H a +
wherein a
d
(3.31)
34
0. Here
iVaO obey the
50
3. FiniteDimensional Localization
fH(Xi), g(X2)},,, which follows from

IH(X2), g(X,)I,,,
(3.39). However,
Theory
=
for
Dynamical Systems
g([Xi, X2])
if there exists
VXI, X2
(3.42)
E g
locallydefined function
a
f such that
ga then
Of
we can remove
0 +
=
1form
df
=
fH a, fj"
the functions 9 a
by the canonical transformation 0 + symplectic potential Of is Ginvariant. Indeed, the
that the
so
Of obeys LVOf
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 Ginvariance (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 Ginvariant and the identity (3.27) relating the Hamiltonians to the symplectic potential by Ha iv0 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
=
iv0 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
2cocycle appearing
in
algebra
H 2(Sl)
=
(3.49)
a
0 for all a, b that the Gaction 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 DuistermaatHeckman Theorem
3.3
51
StationaryPhase Approximation
and the DuistermaatHeckman 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)
nondegenerate 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
1parameter
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

Vdet (x) w
e
iTH(x)
(3.52)
52
3. FiniteDimensional Localization
Theory
for
Dynamical Systems
The partition function determines all the usual the
thermodynamic quantities of energies and specific heats, as well in the canonical ensemble of the classical system.
dynamical system [144],
such
as
its free
all statistical averages However, it is very seldom that
as
one can actually obtain an exact closed partition function (3.52) as the integrals involved are usually rather complicated. But there is a method of approximating the integral (3.52), which is very familiar to both physicists and mathematicians, called the stationaryphase approximation [65, 72, 172]. This method is often employed when one encounters oscillatory integrals such as (352) 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
kth 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 lowestorder
stationaryphase 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 largeT expansion (3.53). The higherorder terms in (3.53) are found from the highermoment Gaussian integrals [157] and they will be analysed in Chapter 7. For now, we concern ourselves only with the lowestorder term in the stationaryphase series of (3.52). where of
Usually
argues that the
one
is minimized
for T
>
dH(p)
=
order of
oo.
phase will
(the ground state)
concentrate around the
points where
However, the localization
is
properly determined by all points where
0 since the contribution from other extrema turn out to be of the
magnitude
as
H
since this should be the dominant contribution
those from the minima
[72].
same
3.3 The DuistermaatHeckman Theorem
53
was essentially born in 1982 general class of Hamiltonian the stationaryphase approximation
The field of equivariant localization theory [39] found a
when Duistermaat and Heckman
systems for which the leadingorder of gives the exact result for the partition function
the
O(I/Tn+l)
the
correction terms in
(3.53)
DuistermaatHeckman 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 fixedpoint set MV of the Taction 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 2form 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

qlt
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 Taction locally on M and the group action preserves the Darboux coordinate neighbourhood. Thus each neighbourhood integration above is purely Gaussian and so all higherorder terms in the stationaryphase evaluation of (3.52) vanish and the partition function is given exactly by the leading term in (3.53) of its stationaryphase 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 DuistermaatHeckman theorem is not that of stationaryphase, but rather of In these coordinates the fiows determined 
2
Of T
course
the
proof
contributing
is
in this
completed by showing case
to
(3.53)

that there is
for details
we
no
refer to
regular
[39].
function of
54
3. FiniteDimensional Localization
the
Theory
for
Dynamical Systems
general localization properties of equivariant cohomology that we Chapter. Suppose that the Hamiltonian vector field V generates a global, symplectic circle action on the phase space M. Suppose further that M admits a globally defined Riemannian structure for which V is Killing vector, as in Section 2.5. Recall from the last Section that the symplecticity of the circle action implies that w + H is the equivariant extension of the symplectic 2form. w, i.e. Dv(w + H) 0. Since integration over the 2ndimensional manifold M picks up the 2ndegree 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 2kform component is a(2k) e iTHWk /(iT)nk 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 skewdiagonal with skeweigenvalues 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 skewdiagonal with skeweigenvalues iA,(p). Introducing the etainvariant 77(7t(p)) of 'H(p), defined as the difference between the number of positive and negative eigenvalues of the Hessian of H at p, i.e. its spectral asymmetry, we find n
,q(li(p))
=
2
sgn
iA,,,(p)
which is related to the Morse index of H at p
by
(3.60)
3.3 The DuistermaatHeckman Theorem
77('H(p)) Using
the
identity
1
eif(111)
=
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) DuistermaatHeckman 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
det(p) aet
(3.63)
(P)
is associated with the
antiselfadjoint
linear operator LV (p) which generates the infinitesimal circle (or torus) action on the tangent space TpM. From the above it then follows that the complex numbers
4(p)
the Cartan
generators)
are just the weights (i.e. eigenvalues of complex linear representation of the circle (or
(3.54)
introduced in of the
action in the tangent space at p and the determinant factors from appear in terms of them as the products
torus) (3.58)
n
e(p)
as
=
(_,),\(p)/2
if each unstable mode contributes
a
(3.64)
At, (p)
factor of i to the
for
integral
Z(T)
fact, dV(p) which appears in (3.58) is none other Pfaff dV(p) (see than the equivariant Euler characteristic class Ev(Yp) critical each of in M normal bundle the of point p E MV. The Arp (2.100)) the Pfaffian Pfaff
above. In
=
the bundle of points normal to the directions Mv, so that in a neighbourhood near MV we can
normal bundle is defined of the critical
point
set
as
write the local coordinates
By
construction,
its
the terms in
(358)
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
TT2n,) U( (.A4).
..'
through nontrivial irreducible representations and we can therefore decompose the normal bundle at p E MV into a direct (Whitney) sum of 2plane bundles with respect to this group action, From
(3.54)
it follows that the induced circle action
on
is
n
Ar,P
NO')
=
P
(3.65)
A=J
is simply Ev (Np(t,)) (3.54) then implies that the equivariant Euler class of N(t') P of the orientation into account JVp induced by the proper iA1, (p) /2. Taking
Hamiltonian vector field
near x
=
p and the Liouville measure, and
using the
56
3. FiniteDimensional Localization
Theory
for
Dynamical Systems
multiplicativity
of the Euler class under Whitney sums of bundles find that the equivariant Euler class of the normal bundle at p is
[21],
we
n
Ev(N(")) P
Ev(A(p)
=_
e(p)
(366)
g=1
which is just the weight product (364). Thus, for Hamiltonians that generate circle actions, the Iloop contribution to the classical partition function (i.e. the DuistermaatHeckman formula in the form
(3.58)) describes the equivphase space with respect to the Hamiltonian circle action on M. The particular value of the DuistermaatHeckman 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 DuistermaatHeckman 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 DuistermaatHeckman 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 nondegenerate 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
nondegeneracy 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 nondegenerate 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
Poincar6Hopf theorem, and employing further orem
(2.103)
we
1: PEMv
with
E(R)
the GaussBonnetChern the
find
sgndet?i(p)
=
(41r)nn!
the Euler class constructed from
f E(R)
(3.71)
M a
Riemann curvature 2form R
M. This relation gives a very interesting connection between the structure of the critical point set of a nondegenerate function and the topology and on
one can also define equivariusing the topological definition of equivariant cohomology [111] which is suitable to the equivariant cohomological ideas that we formulated earlier on. These equivariant generalizations
geometry of the phase
space M. We remark that
ant versions of the Morse and Poincar6 series
58
3. FiniteDimensional Localization
Theory
for
Dynamical Systems
which localize
topological integrals such as (3.71) onto the zero locus of a MathaiQuillen 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 DuistermaatHeckman 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 DuistermaatHeckman integration formula is exact, and are
under the assumptions of the DuistermaatHeckman 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 DuistermaatHeckman 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 stationaryphase 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 
nontrivial
systems will localize in the
sense
of the DuistermaatHeckman theorem. In
Chapter 7 we shall see the direct connection between the higher order terms in the saddlepoint series for the partition function and Kirwan's theorem, and more generally the geometry and topology of the manifold M.
Examples: The Height Rinction
3.5
of We
a
Riemann Surface
some concrete examples of the equivariant localization presented above. One of the most common examples in both Morse theory and localization theory is the dynamical system whose phase
present
now
formalism space is
with g
a
compact Riemann surface Z9 of genus g
'handles')
and whose Hamiltonian hZg is the
(i.e. height
a
closed surface
function
on
Z9
3.5
Examples: The Height
Function of
Riemann Surface
a
59
[29, 85, 111, 153, 157].
For instance, we have already encountered the case S2 in Section 2.1 with the height function h_ro sphere ZO given by (2.1). The symplectic 2form. is the usual volume form of the Riemann
=
wzo
dcos 0 A
=
(3.72)
do
metric g,,, = J,,, of R in 3dimensional 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 DuistermaatHeckman 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 2sphere 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 hVo. Thus, a perfect Morse function
is
Morse indices. Notice that the Hamiltonian vector field V

2ao
here generates an isometry of the standard round metric dO(gdO+sin 20 doodo induced by the flat Euclidean metric of R 3. The differential form (2.123) with this metric is
=
do, which
Now the partition function
expected
as
be written
can
Z_ro (T)
I =

T
f
d
is illdefined 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
pathconnected,
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 socalled classifying bundles of Lie groups (the topological definition of equivariant cohomology) see [111], for example.

60
3. FiniteDimensional 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 simplyconnected phase space (i.e. we
shall
see
the
H,(M;Z)
=
0).
The situation is much different
on
a
multiplyconnected
phase space, which as we shall see is due to the fact that the nontrivial 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 2torus 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 halfplane 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
3space as a doughnut standing on end on xyplane and centered symmetrically about the zaxis. 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)
simplyconnected Riemann surface is a unique complex structure (i.e. a unique way of defin
ing complex coordinates)
treatment
01) COS 02
2form
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+Imrcos0j)sin02
Tr2l =Imrsin0jC0S02 ,z
i
(3.80)
The Hamiltonian
(3.78) has 4 isolated nondegenerate 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 noncontractable loops associated with each Slfactor, implies that the DuistermaatHeckman integration formula should fail in this case. Indeed, evaluating the righthand side of the DuistermaatHeckman formula (3.63) gives eiTh_,l (p)
27ri
e(p)
T
PEMv_,, 27ri T vIIm
=
r
[r
1/2
(1+ e 2iTr2)
+
which for the parameter values iT value 21r
On the other
hand,
e3
an
( v'32
IRe rl112
e
2iTImr
e
2iTIRerl
)1 (3.82)
=
1 and
I + i
r
sinh 3 + 2 cosh 1
explicit evaluation
gives the numerical
(3.83)
1849.33
of the
partition function gives
21r 21r
ZZ, (T)
=
I I dol d02 0
e
iTh_,l (01,02)
0
(3.84)
27r =
27r
e
r2j do, Jo (iT(r,
+ Im
r cos
01))
0
with J0 the
regular Bessel function of order 0 [60]. For the parameter values 2117.13 6 contradicting integration in (3.84) gives Z_ri the result (3.83). Thus even though in this case the Hamiltonian h_rl is a above,
a
numerical
All numerical
software

,
integrations
package
in this Book
MATHEMATICA.
were
performed using the mathematical
3. FiniteDimensional Localization
62
perfect Morse function, space here. This argument
can
it doesn't
connected
sum
for
generate any
be extended to the
Riemann surface
hyperbolic is the gfold
Theory
case
Dynamical Systems
torus action
where the
on
phase
the
phase
space is
a
For g > 1, Z9 Z1# #_T1 of 2tori and therefore its first homology group
Z9,
g > 1
[153].
=
...
is 2g
H, (Z9; Z)
=
(
(385)
Z
i=1
3
as g doughnuts stuck together on end and standing xyplane. 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 nonexactness of the stationaryphase approximation (and even worse the divergence of the stationaryphase 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 noncompact group actions and to noncompact phase spaces are not always immediate [29]. A version of the DuistermaatHeckman theorem appropriate to both abelian and nonabelian group actions on noncompact manifolds has been presented by Prato and of the equivariant localization
Wu in
[141].
assumes
This noncompact version of the DuistermaatHeckman theorem a component of the moment map which is regular and
that there is
bounded from below
(so that the FourierLaplace 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 LiouvilleArnold
integrable
3.6
theorem which is
a
Equivariant Localization and Classical Integrability
generalized,
coordinate
Liouville theorem that dictates when
63
independent version of the classical
given Hamiltonian system will have whose solutions can be explicitly found by integrating a
equations of motion by quadratures [35, 55]. The LiouvilleArnold 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 functionallyindependent and in involution, the action variables
to be
f 1,,, 1, J"'
=
(3.87)
0
and from the Hamilton equations of motion
i,, (t) so
a
canonically conjugated actionangle 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 timeevolution 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 timeindependence or radial symmetry. The symplectic 2form in the actionangle 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 DuistermaatHeckman theorem. The
equations of
motion in this
I,, W where
wl'(I)
tem therefore
=
=
1,, (0)
49I,,H(1).
move
along
global
solutions to the Hamilton
case are
1
00 (t)
The classical
=
00 (0)
+ W/' (I)t
trajectories
of the
the Liouville tori with constant
(3.91)
dynamical Sysangular velocity
64
3. FiniteDimensional Localization
11
=
wt'
(equivalently they
share
Theory
a common
for
Dynamical Systems
period)
which represents
large
a
a
assosymmetry of the classical mechanics. The Hamiltonian vector field ciated with the action variable I,, generates the jLth 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 DuistermaatHeckman theorem. For quadratic functionals of the action variables the associated Hamilto
higherorder
and
general generates a circle action which does not have a angular velocity on the phase space and the DuistermaatHeckman formula will not hold. We shall see, however, that modified versions of the DuistermaatHeckman 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 actionangle 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 wellknown fact that any Hamiltonian system is locally integrable. This is easily seen from the local representation (3.54),(3.55) where we can define p,, I,, sin OP. Then H En=J 12 and 1,, cos OA, q" is
=
V
_
En=J A
a
..
generates
=
rotations of the local coordinate sor

angle variables 0" (rigid locally the metric tendepend only on the action variables
translations in the local
neighbourhood).
components g,,, should be taken
to
Then
radially symmetric in the coordinate neighbourhood). However, local integrability does not necessarily ensure global integrability. For the latter to follow, it is necessary that the neighbourhoods containing the conserved charges I,, be patched together in such a way as to yield a complete set of conserved charges defined almost everywhere on the phase space M. Furthermore, global integrability also implements strong requirements on
I,, (i.e.
g is
the behaviour of H in the vicinity of its critical points. As on, the so
isometry
that the
group of
global
a
we
shall
see
later
compact Riemannian manifold is also compact,
existence of
an
invariant metric tensor in the above for
a
3.6
Equivariant Localization and Classical Integrability
compact phase space is equivalent
global
action of
a
(or
circle
65
to the
requirement that H generates the generally a torus). This means that the Cartan element of the algebra of isometries
more
Hamiltonian vector field V is
a
of the metric g (or equivalently H is a Cartan element of the corresponding Poisson algebra). In other words, H is a globallydefined 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 infinitedimensional 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.
(H11,, 19111 OH)
+
G(I) is an arbitrary function of the action variables. Consequently, in neighbourhood where actionangle 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 2form Kv + f2v introduced in (2.119), we note that f2v is a closed 2form. (but not necessarily non
where a
degenerate)
and that the function KV satisfies
dKv as a
consequence of
(2121) W'
and =
=
(3.96)
ivf2v
(2.118), respectively.
S21"49,Kv V
=
wl"c),H
It follows that
(397)
66
3. FiniteDimensional Localization
and
so
the classical
(M, w, H)
and
equations of motion f2v, (M, Kv) coincide 7
for
Dynamical Systems
for the 2 Hamiltonian systems
,
V' (t) This
Theory
=
Jx4, HJ,,
Jx4, KV JS2,
=
(3.98)
that these 2
dynamical systems determine a biHamiltonian strucare interesting consequences of this structure. The first follows from the fact that if H H(I) as above is integrable, then these actionangle 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 1form 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 biHamiltonian system we can solve explicitly the equivariant version of the Poincar6 lemma. The global existence of the 1forms OF and O(V) is therefore connected not only to the nontriviality of the DeRharn coF homology of M, but also to the nontriviality 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 biHamiltonian structures for a given phase space. Furthermore, it is wellknown that the existence alone of a biHamiltonian system is directly connected to integrability [6, 35]. If the symplectic 2forms w and QV are such that the rank (1, 1) tensor
1form
L is
nontrivial, then
one can
=
Qv w1
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
nondegenerate
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
nonsingular [6].
possibly degenerate
keep
the
on
subman
some
of the
submanifolds,
equations of
motion
3.7
Degenerate
Version of the DuistermaatHeckman 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 LiouvilleArnold 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 DuistermaatHeckman Theorem
Chapter we shall quickly run through some of generalizations of the DuistermaatHeckman theorem which can be applied to more general dynamical systems. The first generalization we consider is to the case where H isn't necessarily nondegenerate and its critical point
In these last 3 Sections of this the
set M V is
now a
submanifold of M of codimension
[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
BerlineVergne 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 nonvanishing in the directions normal to the critical submanifold MV [111]. This defines the normal bundle JVV of MV in M, and the phase space is now locally the disjoint union derivation of the
(3.57).
The Hessian of H
=
M so on
that in
M
a
neighbourhood
near
=
MV IIJVV
MV
we can
(3.104)
decompose the
XA


A X0 + XA 1
(3.105)
where xO are local coordinates on MV, i.e. V(xo) coordinates on Arv. Similarly, the tangent space at any
=
be
local coordinates
as
decomposed
0, and xi are local point x near MV can
as
TxM
=
TxMv G TxNv
TxA(v is the space of vectors orthogonal to those in TxMV. decompose the Grassmann variables qt' which generate the algebra of M as
(3.106)
where
We
therefore
exterior
77
where,q`0
generate the exterior
A
770
+,qA1
algebra AMV andqA1 generate AJVV.
can
(3.107)
68
3. FiniteDimensional Localization
Theory
for
Dynamical Systems
Under the usual assumptions used in deriving the equivariant localization principle, it follows that the tangent bundle, equipped with a LeviCivitaa 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
nontrivial 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 2form 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 2form 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 2form 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 larges exponential in (2.137) using the decompositions (3.115). The Jacobian determinants from the anticommuting and
Degenerate Version of the DuistermaatHeckman 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 larges expansion of the argument of the exponential in (2.137) is given by [121]
nlz
S
S_+00
(S?V)A,(xo)nl L n _L 2
V
2
+ 0 (1 /
I
S00
SnV
VA
I
1
0
(3.116)
"FS)
G'V)1(Xo)(S?V)P'(Xo)X"X' I I +0(11 'Fs) 1A

2
(Qv)O,(xo)R' P(xo)xl'x',q' no
+
expanded the C'functions in (3.116) in their respective Taypoints. Thus with the coordinate change (3.115), the integration over the normal part of the full integration domain
where
have
we
lor series about the critical
A'M
M 0 i.e.
over
(x 1711 )
The result is
an
in
=
(Mv (& A'Mv)
(2.137),
integral
over
(Xv (& A'Arv)
is Gaussian and
(3.117)
be carried out
can
explicitly.
the critical submanifold
27ri
Z(T)
II
d'xo d'770 eiT(H(xo)+w(xo,?7o))
T MvOA'Mv X
Pfaff S?v (xo)
.\/det QV (xo) (MV (xo)
+
(3.118)
R(xo, no))
chv(iTw)jm, Ev (R) JArv
27ri
T MV
where
we
have identified the
equivariant Chern and Euler characters (2.95)
and (2.100) of the respective fiber bundles. In (3.118) the equivariant Chern and Euler characters are restricted to the critical submanifold MV, and the determinant and Pfaffian there
taken
are
over
the normal bundle
jvv. Note Mv is
that the above derivation has assumed that the critical submanifold
connected. If MV consists of several connected components, then the formula (3.118) means a sum over the contributions from each of these components.
large equivariant cohomological symmetry
Notice also the role that the
of the
dynamical system has played here it renders the Jacobian for the rescaling transformation (3.115) trivial and reduces the required integrations to Gaus
ones. This symmetry appears as a sort of supersymmetry here symmetry between the scalar x1' and Grassmann 779 coordinates).
sian
There
are
several comments in order here. First of
of discrete isolated
points,
so
that
r
=
2n, then,
all, if MV
(i.e.
a
consists
since the curvature of the
70
3. FiniteDimensional Localization
Theory
for
Dynamical Systems
normal bundle of
a point vanishes and so the Riemann moment map /IV coincides with the usual moment map dV on TM calculated at that point, the formula (3.118) reduces to the nondegenerate localization formula (3.58) and hence to the DuistermaatHeckman 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 nonsingular symmetric matrix along the directions normal to MV, we can orthogonally decompose the normal bundle,
with the aid of
some locallydefined Riemannian metric on MV, into positive and negativeeigenvalue 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 3space as a doughnut
(3.118) now
sitting on a dinner plate (the xyplane). 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 DuistermaatHeckman theorem to nonabelian 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 GuillemanPrato localization formula reduces the integrals over the dual Lie algebra g* to integrals over the dual of the Cartan subalgebra, using the socalled 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 nonabelian lo
calization formalism and used it to study 2dimensional
YangMills 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
e40aOHa
n!
(3.119)
4
where
as
usual the Boltzmann
map of the Gaction
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 complexvalued 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
Ginvariant
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 translationinvariant
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 2form of M, and performing the Gaussian oaintegrals, 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!
e9
E.(H' )2
(3.121)
72
3. FiniteDimensional Localization
Theory
for
Dynamical Systems
The righthand side of (3.121) localizes onto the extrema of the square of the moment map E.(H')'. The absolute minima of this function are the solutions to H 0. The contribution of the absolute minimum to 0' 0 H' =
=
ZG (the dominant contribution for
s
oo)
+
is
given by
a
[171]
formula
(27r) di G12 vol(G) 'm
ZNin
lim
=
f

soo
IMO
e' +
Mo
where Mo
=
H1(0)IG
(or symplectic quotient)
projection we
that takes
is the MarsdenWeinstein reduced
phase space [97] global minima onto A40 is integration in (3.120) (as one can
X
E H
have assumed that G acts
the over
1
(0)
into its
freely
on
4
H1(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 Gequivariance 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 H1(0) > Mo. The Witten localization formula can in this way be used to describe the
cohomology
given symplectic Gaction
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 DuistermaatHeckman 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
=
soo
1/2
_0 f (27r
es(acos 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 ,,
N1
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]
=
N1
 Pj
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 infinitedimensional functional
trajectories (p(t), q(t)), where for each fixed time slice [0,T], dp(t) dq(t) is ordinary RiemannLebesgue measure. Being an infinitedimensional 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 illdefined 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 socalled 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 RiemannLebesgue 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 piecewiselinear functions.
Note that the transition from the the representation
(i.e.
index).
(4.20),(4.21)
at least be
in terms of
4.1 Phase
If
set q
we
=
q'
and integrate
over
Space Path Integrals
all q, then the lefthand side of
83
(4.20)
yields
i dq (ql eikTlhlq)
eifIT/hll
11
tr
_=
dE
eiET/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 righthand 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 canonicallyinvariant Liouville measure (3.22). Thus the quantum partition function for a generic dynamical system (M, w,
H)
is defined
as
[dAL(x)] e'slxl
Z(T) LA4
[d2nXI
J1 V/det 11w(x(t))JI tE
LM
e'slxl (4.25)
[0,T]
where T
S[X]
f
dt
(0,,(x):V'

H(x))
(4.26)
0
is the classical action of the Hamiltonian
shall set h
=
I for
simplicity, and
system. Here and in the following
the functional
integration
in
(4.25)
we
is taken
4.
84
over
the
x(t)
:
Quantum Localization Theory
for Phase
Space Path Integrals
M, i.e. the infinitedimensional 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 infinitedimensional 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 AtiyahSinger 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 AtiyahSinger 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
evendimensional compact orientable
Riemannian manifold M with metric g of Minkowski signature,
4.2
Example: Path Integral
Derivation of the
iY" Here
y"(x)
49,4
AtiyahSinger
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 spinconnection 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"
143
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
+
spinconnection 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 LeviCivitaChristoffel 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', iy 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 cokernel of the
index(iV )
=
dim ker D

dim coker D
E)
0
0
1
C
=
as
t
(4.31)
0
the difference between the
elliptic operator D,
dim ker 'D

dim ker
Dt (4.32)
86
Quantum Localization Theory
4.
(4.31),
In the chiral representation
the Dirac
SPace Path Integrals
the Dirac spinors, i.e. the solutions Tf of
equation
('Dt Do TTff+ 0
iv Tf are
for Phase
(4.33)
ETI
by their positive and negative chirality spin components P. DTV_ zero mode solutions, E 0, satisfy DtTf+ 0, the index just the difference between the number of positive and negative
determined
Since the
(4.32)
is
=
=
=
chirality zeromode 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 onetoone mapping between positive and negative nonzero 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
trHIIy'
=

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
nonzero
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) 1dimensional 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)
=
trHll(I)' e"11
=
stril e"11
(4.38)
4.2
Example: Path Integral Derivation of the AtiyahSinger Index Theorem
is
topological
a
invariant that is
independent
87
spin and gauge since M is compact by
of the choice of
connections. Here str denotes the supertrace
and,
assumption, there are only finitely many modes which contribute in (4.38). quantities are independent of the parameter T. In the low temperature limit (T + oo) only zero modes contribute to (4.38) according to their chirality. Thus all these
physical relevence of the Witten index is immediate. As the zero engeneral need not be paired, the nonvanishing 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 nonvanishing 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,, dr
=
2
1 +

2
2
S1
(4.40) 4
In
general
an
Ncomponent supersymmetric model
contains N fermion chiral
with 2N associated superpartner bosonic fields Fi, and conjugate pairs N corresponding supersymmetry charges (Qit, Qi) which mix the fields with their
superpartners.
88
Quantum
4.
Localization
Theory
for Phase
Space Path Integrals
01' are the Grassmann (superpartner) coordinates for the particle configurations x ' E M, and we have rescaled the time by T so that time inte
where
grations lie
on
the unit circle S1. Here
V,W"(x(r))
aW"(x(,r))
=
+
r','
(4.41)
along the loop x(r) induced by the Riemannian M, and F,,, 91,A, o%Aj, is the gauge field strength tensor. In (4.40), the particle current :V1 is minimally coupled to the gauge field A/,, and its spinor degrees of freedom couple to the electromagnetic field of A, by the usual Pauli magnetic moment interaction. The action (4.40) has is the covariant derivative connection V
the
on
=

(infinitesimal) supersymmetry Sx"(r)
(4.40)
The action
symmetric model in and superfields. us
briefly
(4.39) (see [71] resented of M is
SV)A(T)
arises from the standard
Zumino) sigmamodel,
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 zeromode 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 nonlinear (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
AtiyahSinger
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 gaugefixed 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 gaugefixed 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 Tindependence and use, in the hightemperature 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 (439). 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 antiderivative 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 Grassmannvalued
anticommutator
[ 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
=
dr
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
dr
01' (r)
6
(4.49)
6xll(r)
S1
(4.48)
that
we see
be
can
expressed
S?[x, 0]
as a
=
Dexact
quantity
DZ[x, 0]
(4.50)
where Z [X,
V)]
=:idr
A,,(x(r))} 0"(r)
+
=i
S1
dr
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 'presymplectic' structure because although it is Dclosed, Db[x,01 0, it is not necessarily nondegenerate on the loop space. It is this interpretation of supersymmetric theories in general that makes infinitedimensional 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
=
=
dr b "
1
(r)6 6
(4.52)
V) A (r)
S1
and define the
Dj
=
corresponding graded equivariant
D +
1,b
(01' (r)
dr
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 dr
S1
A
(r)
6 
JxA(r)
+
0 11 (T)
6
To _('r)
)
(4.53)
as
the action of
5011(r)
) =j S1
d'r
is the generator
d
77
(4.54)
Example: Path Integral Derivation of the AtiyahSinger Index Theorem
4.2
so
that its action
D?W[x,01
on a
loop d
dr
=
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 wellknown 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
dr
(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 illdefined 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 semiclassical approximation is exact. The Tindependence can be understood from the point of view that if we differentiate the righthand 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 Wittentype topological field theories) in general [22]. The above connection between the formalisms of the previous Chapters and the AtiyahSinger index theorem is the usual intimate connection between standard supersymmetric invariance of BRSTexact
models
(for
instance those which arise in the DuistermaatHeckman
tation of the quantum mechanics of spin
[26].
[156, 4])
and
interpreequivariant cohomology
92
Quantum Localization Theory
4.
for Phase
Space Path Integrals
We now use the fact that (4.57) can be evaluated for T * oo and use a trick similar to that in Section 3.7. We introduce based loops on LMOLAIM,
X" (r) with
(X0, ?PO)
X, JP (r) 0 +
=
M(&AIM
E
01(r)
by6
[d2nX] [d
+
000
the constant modes of the fields and
constant fluctuations about these measure
b'0
=
2n
(4.58) the
non
zeromodes, and define the path integral
11
d2nXO d2nV)O
d2n,
(r)
d2n (,r)
(4.59)
rES1
We then rescale the nonconstant 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
nonconstant 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
AtiyahSinger 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 nonconstant modes of x(r)). Thus (4.62) coincides with a formal application of the degenerate DuistermaatHeckman integration formula (3.118) (more precisely the degenerate version of the BerlineVergne theorem) to the infinitedimensional 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 infinitedimensional determinants
[1021.
Notice first that here the infinitedimensional 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 finitedimensional 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 infinitedimensional Pfaffian (see (362)) has to be regulated and is given by the AtiyahPatodiSinger etainvariant [41, 158] of the Dirac operator 0, R, 
R)
=
lim s+O
t
dA
sgn(A)IAI'
+ dim
ker(a,

R)
CO
=
Jim 80
f
I
_r
where the integration and
(8+21)
dt
t(s1)12
(and/or sum)
is
over
tr
all
(c),

R) et(a,
nonzero
R)21 1
eigenvalues
A of
(4.63)
'9,

R
00
J
F(X)
dt
tx1 et
X
> 0
(4.64)
0
is the Euler
Next,
gammafunction.
we
ularizations
evaluate the determinant
using standard supersymmetry
reg
[5, 48]
for firstorder differential operators defined on a circle. The most convenient such choice is Riemann zetafunction regularization. The nonconstant singlevalued eigenfunctions of the operator '9, on S' are
e2,jrikr ,where k
are nonzero integers. Since the matrix R is antisymmetric, skewdiagonalized into n 2 x 2 skewdiagonal 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 nonzero
=
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 etainvariant
(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
)10
We find that
=
(21ri )4
(O)
=
(27r 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 Agenus (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 AtiyahSinger index theorem for a twisted spin complex [41]. We see then that a formal application of the BerlineVergne theorem yields the wellknown AtiyahSinger 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
nonabelian 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 GaussBonnetChern theorem. An equivariant generalization then yields the Poincar6Hopf theof classical Morse theory [166]. We shall elaborate on some of these ideas, as well as how they extend to infinitedimensional cases relevant to topological field theories, in Chapter 8. Finally, we point out that the equivariant cohomological interpretation above is particularly wellsuited to reproduce, the CalliasBott index theorems [70], i.e. the analog of the AtiyahSinger index theorem for a Dirac operator on orem
an
is
odddimensional noncompact 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
infinitedimensional,
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 higherdimensional AtyahSinger 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, infinitedimensional 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 infinitedimensional 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
infinitedimensional nonabelian
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 infinitedimensional 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 anticommuting periodic paths 77A(t) which generate LAM and which loop space 1forms. With this, we
to be identified
are
can
define
loop
as
the basis
dx"(t)
of
space differential kforms
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 2form
is determined
by
a
loop
space sym
T
dt dt'
S? 0
1
S?,,, [x; t, t],q" (t),q' (t')
(4.74)
Loop Space Symplectic Geometry and Equivariant Cohomology
4.3
97
which is closed
dLO or
x"(t)
in local coordinates
j JXA (t) Thus
S?"\ [X; tt, t// ]
j
Q'\j, [X; tf, t// ]
JX" (t)
(4.75)
0
LM,
on
6 +
=
+
6XII W
S?,,, [x; t', t"]
=
0
(4.76)
apply the infinitedimensional version of Poincar6's lemma to locally in terms of the exterior derivative of a loop space 1form
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
nondegenerate,
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
infinitedimensional spaces of sections of the
pullback
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 S2 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 canonicallyinvariant closed form products of S? with itself,
[d[tL(X)l The
loop
:
on
LM given
[d2nXj V/det 11f2j[
space Hamiltonian vector field associated with the action
has components
(4.81)
(4.26)
98
Quantum Localization Theory
4.
for Phase
Space Path Integrals
T
Vs, IX; t]
=f
dt' f2l" [x; t, t']
bA W
=
Jxv (t,)
V"(X(t))

(4.82)
0
with VA of
=
wl"o%H
as
usual the Hamiltonian vector field
on
M. The
zeroes
Vs LMs
are
the extrema of the action
of the
(4.83)
E
(4.26)
and coincide with the classical trajectories
LM:
Vs[x(t)]
01
Jx(t)
=
=
i.e. the solutions of the classical Hamilton
dynamical system, loop space
of motion. The
Wl'[x; t]
vector field
is
contraction operator with
respect
to
a
equations
loop
space
given by T
iw
=
I
dt W" [x;
6
t]
(4.84)
J771, (t)
0
Thus
we can
define
a
loop
space
exterior derivative
equivariant
Qw whose square is the Lie derivative
(4.85)
dL+iW
=
along
the
space vector field
loop
W,
T
Q2W
=
dLiW
+
iwdL
=
dt
fw
(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 finitedimensional case using the functional Berezin integration rules to absorb the determinant factor into the exponential in terms of the anticommuting 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 superloop space LM (9 LA'M. From this we can now formally describe so
the
SIequivariant 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
1form 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
0form 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 finitedimensional
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 infinitedimensional manifolds. Nonetheless, the localization priniciple can be established by interpreting the equivariant cohomological structure on LM as a "hidden" supersymmetry of the quantum theory. In this way
function
100
Quantum Localization Theory for Phase Space Path Integrals
4.
has
one
sort of Stokes' theorem in the form of
a
with this supersymmetry
(as
was
the
identity
Ward
a
in Section 4.2
case
associated
above),
where
we
interpret the fundamental localization property (2.126) as an infinitesimal change of variables in the integral. The partition function (4.87) can be interpreted as a BRST gaugefixed 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 evendegree, commuting loop space forms (bosons) into odddegree, anticommuting 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 (nonsupersymmetric) quantum theory as a supersymmetric or topological field theory. It was Blau, KeskiVakkuri 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 gaugecovariant 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
superJacobian
on
Principle
101
space LM 0 LAM whose corrections
superloop
the
Localization
related to anomalies and BRST supersymmetry breaking. The arguments below are therefore valid provided that the Qssupersymmetry above is not are
broken in the quantum theory. The argument for infinitedimensional localization Consider the 1parameter
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 1parameter family of gauge fermions with 0 and 0,=o 0). As in the finitedimensional 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
=
BRSTinvariant.
However, the corresponding superJacobian arising in the Feynman in (4.96) on LM 0 LAIM is nontrivial and it has precisely the
sure
functional form tinent
as
that in
superJacobian
standard BRST transformation
a
here is
[1181.
measame
The per
given by the superdeterminant J.T 6x,;
[d2nt] [d 2nq]
Tx =
sdet
57)
jq jq
[d 2nx] [d
2n
(4.99)
Tx TO and the
path integral (4.96) is invariant under arbitrary smooth changes J,\, the identity
of
variables. For infinitesimal
tr
log 11AII
=
log det 11AII
(4.100)
102
4.
Quantum Localization Theory
implies that the superdeterminant
for Phase
in
the supertrace, the superloop 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,,]

e5A*QsP[d2n x] [d2n,,] (4.101)
substituting the change of variables (4.98) with superJacobian (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 Aindependence (or the V)independence more generally) of (4.96) is a specialization of the FradkinVilkovisky theorem [13, 14, 118] to the supersymmetric theory above, which states that local supersymmetric variations of gauge fermions in a supersymmetric BRST gaugefixed path integral leave it invariant. Indeed, the addition of the BRSTexact term QSO can be regarded as a gaugefixing 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 BRSTexact 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 BRSTexact path integrals, also known as Wittentype 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 A00
[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
metricdual form T
0
=
f
G,,, [x; t, t'] W1 [x; t],q'(t')
dt dt'
(4.104)
0
0 is loop space vector field W. The supersymmetry condition Lso additional the and 0 the to requireLsG Killing equation equivalent
of the then ment
=
=
LsW
=
0
on
W
1,
where T
'CSW
=
I (Tt 'CV(X(t))) d
dt
W [X;
_
(4.105)
t]
0
In
principle
but
we
there
are
many useful choices for W
shall be concerned
mostly with those
W" [x; where the parameters r, scheme.
s are
t]
=
r.V'(t)

obeying such
which
can
a
restriction,
be summarized in
(4.106)
sV" (x (t))
chosen appropriate to the desired localization
As for the metric in (4.104), there are also in principle many possibilities. However, there only seems to be 1 general class of loop space metric tensors to which general arguments and analyses can be applied. To motivate these, we note first that the equivariant exterior derivative QS can be written as
QS
=
Q

iv
=
dL
+
ii

(4.107)
iV
and the square of the operator Qj, is just the generator of time translations T
T
Q?x
=
dt
C,
(V'(t) 6XIL(t)
+
"(t)
6
JWIM
we assume
that the
We also
require that
d
Tt
(4.108)
loop space Hamiltonian vector parametrized by a parameter
field generates an S'flow on loop space, ,r E [0, 1] so that the flow is x11 (t) + x1'(t;,r) with x" (t; the
9
dt
0
0
This operator arises when
f
the combination
(4.104)
0)
=
x1' (t;
1),
be such that it determines
such a
ho
topological motopically effects into the path integral (4.1,03) when evaluated on contractable loops. For the most part, we shall be rather cavalier about this requirement and discuss it trivial element
as
above,
only towards the end of this Book.
so
that it introduces
no
extra
104
4.
Quantum Localization Theory for Phase Space Path Integrals
loop space coordinates xA(t) the flow parameter loop (time) parameter t  t +,r. In this case we have
that in the selected shifts the
VS11 IX; t] '9xA(t;'r) t9r and the supersymmetry transformation
Q.+x11(t) =,q"(t) which
we
above. In
Ir=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 singlevalued functionals on LM, i.e. 0(0) O(T). This form of the U(1)equivariant the modelindependent circle action. called the is loop space cohomology on and the
(4. 111)
topological
invariance of the quantum by elements
under BRSTdeformations
=
We shall therefore demand that the localization functionals in
(4.104)
be
modelindependent Slaction 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 ultralocal 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]
(4113)
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 (4113), 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
globallydefined 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 infinitedimensional 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
(236)
satisfied
(as
for the finitedimensional
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 DuistermaatHeckman integration formula, whose derivation follows the loop space versions of the steps used in Sections 2.6 first infinitedimensional localization formula that
and 3.3. We
assume
that the action 5 has that the
(finitelymany) 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
secondorder variation of S is
Under these assumptions,
we
nonvanishing
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 finitedimensional integral (4.103) gives
localization
case, the evaluation of the
106
4.
Quantum Localization Theory for Phase Space Path Integrals
f [d2nx] VFdet 11011 Vdet_JJJVSJJ J(VS) e slxl
Z(T)
LM
det 11 Jl',9t
[d2nx] Vdet 11 Q 11
a,

V
(4.115)
LM
w"'aH) e'slxl
x
_11w(x(t)) 11 ,Fdet
e'slxl
x(t)ELMs will be used following the symbol the determinants which into prefactors
where here and in the
absorption of infinite

to
signify
the
arise from the
functional Gaussian integrations. The functional determinant in the denom(4.115) can be evaluated in the same manner as in Section 4.2
inator of
above with R
+
odic functions
on
2.7rik
>
Agenus
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
(4116)
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 cnumbers as in the classical theory. For h * 0 we can naturally evaluate the path integral by
stationaryphase 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 oneloop determinant factors in (4.115)) around these classical trajectories.
4.6 The NiemiTirkkonen 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 socalled 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 HamiltonJacobi 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 nonvanishing 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 semiclassical 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 NiemiTirkkonen 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 Tperiodic time T. of the values discrete for propagation eral a nonisolated 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 nonconstant
=
=
=
=
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 2form 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
DuistermaatHeckman formula of Section 3.7.
=
=
of this later
on.
analog of the degenerate We decompose LM and LA'M
space
Quantum Localization Theory for Phase Space Path Integrals
4.
108
into classical modes and fluctuations about the classical solutions and scale
the latter
by
X1, (t)
:t(t)
V, (t) + X, (t) /,V"Af
LMs '(.t(t))=XtV'
where
Vs
=
11VA,
E
kernel of the
loop
are
q '(t)
'(t)
=
(4.117)
f
the solutions of the classical equations of motion, i.e. 0, and 11(t) dV'(t) E AlLms span the =

space Riemann moment map,
(S?S)4(t)qV
=
(4.118)
0
where T
Ds
=
J
dLO
dt
6 JXA
(g Vx vs, ) n IIn
(4.119)
V
0
0 given they obey
with i.e.
in
(4.114).
particular, this implies
In
(JI, at

V
a, V1,(,t)) qV
(4.117) obey q f'(T) 0.
The fluctuation modes in
x" (T) f
=
The
that
q1' (t)
are
Jacobi
fields,
the fluctuation equation
0 and
qf1'(0)
superloop
[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 finitedimensional 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 2form of the g metric g evaluated on LMS. In (4122) the Pfaffian is taken over the fluctu
=
ation modes

xl'(t) f
the normal bundle
about the classical trajectories tA(t) E LMs (i.e. along A(LMS in LM), and the measure there is an invariant
measure over the moduli space of classical solutions which is itself a symplectic manifold. The localization formula (4.122) is the loop space version of the
degenerate localization formula (3.118)
in which the various factors
can
be
4.6 The NiemiTirkkonen Localization Formula
109
interpreted as loop space extensions of the equivariant characteristic classes. particular, in the limit where the solutions to the classical equations of motion VA(x(t)) 0 become isolated and nondegenerate 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, Tdependent structure, i.e. it is usually a highly nontrivial problem to solve the classical equations of motion for the Tperiodic 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 (4106) consists of the constant loops points on M, so that the canonical localization integral will reduce to an integral over the finitedimensional 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.
righthand 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 4m (t)
f
dt
e(t)
=
0
0
Some features of the space of Tperiodic classical trajectories for both energy conserving and nonconserving Hamiltonian systems have been discussed recently
by
Niemi and Palo in
[119, 1221.
110
Quantum Localization Theory
4.
decomposition (4.124) is essentially some complete sets of states JX1J'(t)JkEZ k The
(t)
k
k
Space Path Integrals
for Phase
a
Fourier
and
(t)
Feynman
so
in terms of
that
Uk"77k"(O
M
k560
and the
decomposition
f?7A(t)JkEZ) k
(4.126)
kOO
path integral
in the
measure
is then defined
just
as
before
as
[d 2nx] [d 2n.1
=
2n
d
xo d
2n
11
77o
d 2n,
(t) d2n (t)
tE[O,T] =
d
2n
xo
d2n 77o
11 d
2n
Sk
(4.127)
d2n Uk
kOO
resealing
With the
QSIP
(4.124)
in
of the fluctuation
modes,
the gauge
fixing
term
is T
QsV)
=
I 1 dt
x
it
((S?V)P'Vat
_
g,,,at2) Xv +
1Rpvxl'. bv +
2
Agjv9t 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 Tdependent 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 eiTH comes from evaluating the action S[x] on the constant loops x E M C LM, so that formula
to the
4.6 The NiemiTirkkonen Localization Formula
(4.129)
is another sort of semiclassical 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 equivariantAgenus (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 (finitedimensional) integral over the phase of equivariant characteristic classes in the U(1)equivariant coho
This is the NiemiTirkkonen 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
timeindependent 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 AtiyahSinger index density of a Dirac operator with background gravitational and gauge fields, and it therefore represents a sort of equivariant generalization of the AtiyahSinger 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 lefthand 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 AtiyahSinger index for =
twisted spin complex (the 'twisting' here associated with the usual sYmplectic line bundle L * M). Thus the localization formalism here is just a
112
Quantum Localization Theory for Phase Space Path Integrals
4.
general case of the localization example of Section 4.2 above which reproduced quite beautifully the celebrated AtiyahSinger 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 AtiyahSinger
topological
field theories

invariants
nature of
Section 4.10
index.
4.7 Connections with the DuistermaatHeckman
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 DuistermaatHeckman 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 finitedimensional 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 AtiyahSinger 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 finitedimensional localization formula (3.118) only equivariant Agenus which arises from the evalua
Hamiltonian gives
in terms of the
onto
Mv
a
localization of the
Chern class restricted to
equivariant class and Agenus
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 DuistermaatHeckman
Integration
Formula
113
quite natural according to the general supersymmetry arguments above (as DiracAgenus 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 NiemiTirkkonen 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 nondegenerate 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
rTetl
det
o9t
f2v 
Qv
eiTH(p)
J(V) PEMv
v det_Qv
A(TQV)
(4.135) (ordinary) Dirac Agenus 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 timeindependent at
a

classical trajectories of the dynamical system, yielding a localization formula that differs from the standard DuistermaatHeckman 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 nonintersecting and each classical trajectory can be
contracted to
(for
a
critical
point of H through
instance when H1 (M;
boundary
condition
xl'(0)
R)
=
a
family
of classical
trajectories
0), and that the period T is such that the x" (T) admits only constant loops as solutions to =
the classical equations of motion. The localization onto the critical points of the Hamiltonian is not entirely surprising, since as discussed at the beginning
114
4.
for Phase
Quantum Localization Theory
Space Path Integrals
of the last Section for Hamiltonian circle actions
MV
and
coincide.
(4.130)
dex
is
in this
particular,
tonian H is
[85].
We
see
arguments
a
that the
conclude from Kirwan's theorem that the Hamil
case
perfect
Morse function that admits
(such
as
Kirwan's way that
exactly phase space integrals.
4.8
locuses LMS
zero
of
only
even
Morse indices
therefore that localization formulas and various Morse theoretic
the
in
M the
from the
Drawing
DuistermaatHeckman theorem in
on
analogy (4.135) with the equivariant AtiyahSinger 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 finitedimensional 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
coordinateindependent parameters. The transforto the leading order infinitesimal limit of
(4.137),(4.138) corresponds
mation
the canonical transformation
x14
*
f ec"., 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 timedependent 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 nontrivial 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 timedependent 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 11011
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 NiemiTirkkonen formula
to arrive at
(iTH 1i ajol) f chi,Ia (iTw) AAj,.i.(TR) 1

exp
M
(4.142) JO=O
and so the path integral now localizes to a derivative expansion of equivariant characteristic classes. The localization formula (4.142) is valid for any inte
grable Hamiltonian system whose conserved charges joaja generate a global isometry on M, and consequently the localization formalism can be used to establish the exact quantum solvability of generic integrable models.
Quantum Localization Theory for Phase Space Path Integrals
4.
116
examples of integrable models where valid, and this has led (4.115) to the conjecture that for a large class of integrable field theories a "proper" version of the semiclassical 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 nontrivial
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 righthand 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 nonintegrable 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 semibounded 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 eiT.F(H) 11 which is not a tempered distribution and thus eliminating any rigorous attempts to make the path integral a welldefined 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 modelindependent 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 BRSTexact, 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 modelindependent S' loop space equivariant cohomology which renders their actions BRSTexact. 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 nonlinear functional of 0 is just the Nicolai transformation in supersymmetry theory [22, 116), i.e. the
118
Quantum Localization Theory for Phase Space Path Integrals
4.
change into
a
of variables that maps the bosonic part of the supersymmetric action change of variables coincides
Gaussian such that the Jacobian for this
with the determinant obtained of the
supersymmetric
struct
a
by integrating
over
the bilinear fermionic part
action. This observation enables
one to explicitly conpath integral (4.145). Notice that when F(H) is either linear or quadratic in the observable H, the Nicolai transform is directly related to the functional Fourier transfor
localization for the
F(H),
mation of
T
T
exp

if
f [do]
F(H)
dt
i
exp
dt
P(O)
i
exp
0
LR
0
f
T
f
dt
OH
0
(4.147) However, for more complicated functionals F(H) this straightforward. In particular, if we change variables
(4.146),
sian transformation
we
i
f
in the Gaus
find
T
exp
connection is less
T
dt
97(H)
f [do]
=
LR
0
fj
'(O)
if ( dt
exp
tE [0,T]
1 2
2 (0)

OH)
0
(4.148) that the effect of this transformation is to isolate the isometry generator H and make it contribute linearly to the effective action in (4.145) (as we
so
did in the last
Section).
prescriptions of Section
Substituting (4.148)
This allows
one
to localize
(4.145) using
the
general
4.4 above. into
(4.145),
we
then carry out the
led to the NiemiTirkkonen localization formula
same
steps which
(4.130). However,
now
there
auxilliary, timedependent 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 BRSToperator incorporating the supermultiplet (0,'q) such that the partiis
an
tion function is evaluated with
the
BRSTcomplex
of
physical
BRSTexact action whose argument lies in states and, as was the case in Section 4.2, the a
NiemiTirkkonen localization onto constant modes becomes manifest. This is the socalled 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
BRSToperator
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 semibounded 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 NiemiTirkkonen localization H 2, we find 0( ) portant special case.F(H)
(4.149)
1 and
(4.149)
formula.(4.130). (i.e. F(O)
=
localization formula
=
=
reduces
In the im
02)
and the
becomes
00
Z(TIH 2)
f doo
_
e
iT020 /2
v
( iTw)
Ao. v (TR)
A
(4.150)
M
00
which is the formal
f choo
path integral generalization of the Wu localization formula
(3.128). In fact, the above dynamical treatment of the multipliers 0 suggests a possible nonabelian 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 nonabelian Hamiltonian moment map (330) 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 timedependent 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 Gequivariant cohomology of M in the Weil algebra using the BRST formalism [129, 161] (see Appendix B). When the 0' are fixed we assume
parameters, the action functional
(4.26) generates the
action of
S'
on
LM in
the model
independent circle action described in Section 4.4 above. However, when the 0' are dynamical quantities, S generates the action of the semidirect
the
product LGDS', where the
loop parameter
t and LG
group G. These actions
are
action of
C' (S',
G)
S' corresponds
to translations of
loop group of the isometry generated, respectively, by the loop space vector =
is the
120
4.
Quantum Localization Theory for Phase Space Path Integrals
fields T
dt
Vsi
:V'(t)
JX/1 (t)
0
T
I
VLG
T
0'(t)w1"(x(t))
dt
(
j Jx (t)
H')
j 
=
Jxt' (t)
0
j
dt
0'(t)V'(t)
0
(4.151) The commutator
algebra of the
(4.151)
vector fields
is that of
LG2DS'
on
LM, T
VS1, VLGI
dt
'H
[Va (t), Vb (tl)]
a
=
fabc Vc(t)J(tt) (4.152)
0
The equivariant extension of the
symplectic
2form 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
finitedimensional 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 semidirect product action of LGs)Sl on LM (the nonabelian 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 NiemiTirkkonen localization, and it yields the localization formula [161]
such that the a
quantity S + Q
nonabelian
=
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
nonabelian version of the
Topological Quantum
4. 10
presented
Field Theories
121
12
in Section 3.8
Notice that the primary difference between this nonabelian 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 BRSTexact 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
righthand
zero
=
(4.115)
lx=xo
(4.155)
M
and only the zero modes of the symplectic 2form 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
AtiyahSinger 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 equivariantlyexact action (4.91). Note that this is precisely the solution to the problem of solving is
sentative of the manifold M which contains
The
procedure outlined
above could also be
a
no
employed
in the discussion of the
Witten localization formalism in Section 3.8. This has been
carrying
out the
implicitly
equivariant integrations there (see Appendix B).
done in
122
4.
Quantum Localization Theory for Phase Space Path Integrals
loop space equivariant Poincar6 lemma for S + 0. If we assume that the symplectic potential is invariant under the global U(I)action on M, as in (3.26), then the Hamiltonian is given by H iVO and the loop space 1form ,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 1form (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 Aindependence of the arguing the of Consider symplectic 2forms 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 2surface 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
WessZuminoWitten
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 nontrivially on the localization 1form 0 =,O and thus also on the action S. Thus the path integral (4.157) defines a consistent quantum theory only when the symplectic 2form (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 (4161)). 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 NiemiTirkkonen localization formula, the BRSTexact 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 Qbexact piece of this action corresponds to the AtiyahSinger 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 ivexact 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
=
=
Gindex)
indeXH(iY; T) generated by
=
lim A
trJJ
e
iTH
( eAl)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) AtiyahSinger 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 righthand side of the Lefschetz number (4.162) is independent of A c: R+, and, therefore, when either DtD or VDt has no
124
4.
zero
i)F
Quantum Localization Theory
modes,
we can
contribute to
take the limit A
for Phase
+
(4.162). Consequently,
SPace Path Integrals
0 there and
the
only
the
zero
modes of
equivariant index coincides with
the character
indeXH (iV; T) of the
irreducible) representation by the zero modes of i)F.
(reducible
G determined
strR
=
or
e
iTH
(4.163)
R of the Cartan element H of
LVO Consequently, in the case of Hamiltonian systems for which LVg 0, the NiemiTirkkonen 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 infinitedimensional cases. Note also that the path integral (4.157) has the precise form of a Wittentype or cohomological quantum field theory, which is characterized by a classical action which is BRSTexact 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 semiclassical 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 Ginvariance 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 BRSTexact, globally
the quantum
theory
is nontrivial
and may not be given exactly by a semiclassical approximation. Then the path integral (4.96) has the form of a gaugefixed topological field theory,
Schwarztype 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 2form 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
( +dLfli 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
equivariantlyexact The nontriviality
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 nontriviality 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
actionangle 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 actionangle 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 multiplyconnected 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 WessZuminoWitten 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 multiplyconnected 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 4dimensional YangMills can be represented in terms of a locally exact form and is therefore
which
nontrivial
only
for
spacetimes which have noncontractable 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 finitedimensional localization theorems, but within the loop space localizaframework, where the arguments for localization are based on formal su
tion
persymmetry arguments on the infinitedimensional 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 noncompact group actions. For instance, if we wanted to apply the localization formalism to an ndimensional R 2n,then we potential problem, i.e. on the noncompact phase space M would expect to be allowed to use a Hamiltonian vector field which generates noncompact global isometries. As we have already emphasized, the underlying feature of quantum equivariant localization is the interpretation of an equivariant cohomological structure of the model as a supersymmetry among the physical, auxilliary or ghost variables. But as shown in Section 4.3, this structure is exhibited quite naturally by arbitrary phase space path integrals, so that, under the seemingly weak conditions outlined there, this formally results in the equivariant localization of these path integrals. This would in turn naively imply the exact computability of any phase space path integral. Of course, we do not really expect this to be the case, and there is therefore the need to explore the loop space equivariant localization formalism in more detail to see precisely what sort of dynamical systems will localize. In this Chapter we shall explore the range of applicability of the equivariant localization formulas [40, 159] by presenting a more detailed analysis of the meaning and implications of the required localization symmetries, and we shall work out numerous explicit mathematical and physical applications of the formalisms of the previous Chapters. As we shall see, the global isometry condition on the Hamiltonian dynamics is a very restrictive one, essentially meaning that H is related to a global group action (2.36). The natural examples of such situations are the harmonic oscillator and free particle Hamilto2n nians on R (the trivial Gaussian, free field theories), and the quantization of spin [117] (i.e. the height function on the sphere), or more generally the quantization of the coadjoint orbits of Lie groups [4, 26, 85, 128, 156, 159] and =
R. J. Szabo: LNPm 63, pp. 127  201, 2000 © SpringerVerlag Berlin Heidelberg 2000
128
5.
Equivariant Localization
on
Simply Connected
Phase
Spaces
the equivalent KirillovKostant geometric quantization of homogeneous phase space manifolds [2, 3]. Indeed, the exactness of the semiclassical 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 grouptheoretical 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 DuistermaatHeckman
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 2dimensional 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 4manifolds), so
that
a classification scheme such as the one that foll6ws does not generalize higherdimensional models. We shall, however, illustrate how these situations generalize to higher dimensions via some explicit examples which
to
will show that the 2dimensional classifications do indeed tell
us
about the
properties of general localizable dynamical systems. In particular,
we
shall
5.
see
Equivariant Localization
on
Simply Connected
Phase
Spaces
129
that from certain points of view all the localizable Hamiltonians repre"generalized" harmonic oscillators, a sort of feature that is anticipated
sent
from the
previous integrability arguments and
the local forms of Hamiltoni
which generate circle actions. These seemingly trivial behaviours are a reflection of the large degree of symmetry that is the basis for the large reans
duction of the
analyse
complicated functional integrals
to Gaussian
ones.
We will also
in full detail the localization formulas of the last
Chapter, which will therefore give explicit examples of the cohomological and integrable models that appear quite naturally in (loop space) equivariant localization theory. This analysis will also provide new integrable quantum systems, as we shall see, which fall into the class of the generalized localization formulas (e.g. the NiemiTirkkonen formula
(4.130)),
but not the
approximation. Such examples represent
more
traditional WKB
major, nontrivial 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 finitedimensional (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 finitedimensional and loop space localization formulas. Furthermore, because of their classical properties, coherent states are particularly wellsuited for semiclassical studies of quantum dynamics. We shall see that all the localizable dynamical systems in 2dimensions 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 simplyconnected 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 noncompact) semisimple 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)
Vy
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'(g1yg)
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 Ginvariant
henceforth that H1 (G)
symplectic
structure
which is defined
the point A E
g*
is
by the KirillovKostant given by WA
where T is
a
2
A T
on
2form
the
[2, 3].
algebra
H2 (G)
coadjoint
=
0.
orbit
This 2form at
A, T])
1form with values in the Lie
=
(5.3) g which satisfies the
equation
dA(y)
=
ad* (T)A(y)
=
A([I, 7])
Vy
E g
(5.4)
5.1
and
ad*(T)
Coadjoint Orbit Quantization and Character Formulas
131
denotes the infinitesimal
The 2form
(5.3)
is closed and
coadjoint action of the element T E g. nondegenerate on the orbit (5.1), and by OA, by symplectic (canonical) transforma
construction the group G acts on respect to the KirillovKostant 2form. 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 finitedimensional 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,
antiholornorphic
(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 antiholomorphic 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
qALa2A
:
A(p,q) MG A(p,q+l) MG (qT' d.P). The antiholomorphic exterior derivative 0 is called the Dolbeault operator, and the nilpotency of d now translates 
into the set of conditions

132
Equivariant Localization
5.
=,92
0
Finally,
let
fined rank
T(0,1)MG)
us
note that
(1, 1)
Simply Connected
on
=
02
a6
=
Phase
Spaces
0,9
+
(5.8)
complex manifold always possesses a globally de(i.e. an endomorphism of the space T(',O) MG 6) It can be defined locally by
a
tensor field J
with j2
=
_1.
J'A
jr"a
ij/1 V
=
i jrA,
(5.9)
with all other components vanishing, and it is known as a complex structure. Given this important property of the coadjoint orbit, we now introduce local complex coordinates (zA,,P') on Mc which are generated by a complex
(5. 1)
0 and topological features H1 (M G; Z) dim HC is the rank of G and Z' r corresponds to the lattice of roots of Hc [162]. The cohomology classes in H 2(MG; Z) are then represented by r closed nondegenerate 2forms 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
nondegeneracy 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 2forms
can
antiholomorphic 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 DolbeaultGrothendieck lemma. Since the 2forms w(') in the case at hand
The
are
closed under both c9 and
that
locally they
can
be
6,
the DolbeaultGrothendieck lemma
expressed
w(')
in terms of C'functions
or
=
F(')
On
implies MG as
06F(')
(5.13)
in local coordinates
W
4AZ" O In
general,
is called
a
a
complex manifold
=
192 F(')(z,.
)
(5.14)

azt'09' Fl with
a
symplectic
Kdhler manifold. The closed 2forms
structure such
(5.10)
are
as
(5. 10)
then refered to
as Kiffiler classes or Khhler 2forms, the associated metrics (5.11) are called Khhler metrics, and the locallydefined 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 Gaction 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 Ginvariant integral symplectic 2 structures on MG Since H (G) 0, the 2cocycles in (3.39) vanish and this Gaction determines group isomorphisms into the Poisson algebras of M. This also follows directly from the property (5.5) of the KirillovKostant 2form above. Notice that the only nonvanishing components (up to permutation of indices and complex conjugation) of the Riemannian connection
so
that the natural
Consequently,
symplectic
action of G
=
the closed 2forms
=
(5.11)
and curvature associated with the Kdhler metric
r"'A
=
RA
gAP'9Vg'X'0
g
=
49IA 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 onedimensional [53, 162]. In this basis, the generators have the nonvanishing
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 Ginvariant symplectic 2form Cartan
=
,
g0C
=
=
.
.
.
,
I
134
5.
Equivaxiant Localization
on
Simply Connected
Phase
Spaces
r
w(A) The
symplectic potentials associated
O(A) To construct
=
E ( \i
i9F(') azil
Aiw(') with
dz"
(5.20)
(5.20)
are
i9F(') 
a
A
+ dF
(5.21)
topological path integral from this symplectic structure, we a Hamiltonian satisfying (3.27), i.e. a Hamiltonian which is given by generators of the subalgebra of g 0 C which leave the symplectic potential (5.21) invariant. These are the canonical choices that give welldefined functions on the coadjoint orbit (5.1). As remarked at the end of Section 3.2, there usually exists a choice of function F(z,, ) in (5.21) for which this subalgebra, contains the Cartan subalgebra h 0 C of g (9 C. Let a
need to construct
H(A)
be the generators of h 0 C in the representation with
highest weight
vector A. Then the Hamiltonian
H(A)
hiH A)
(5.22)
satisfies the
required conditions and the corresponding path integral will adtopological form (4.157). Note that this is also consistent with the integrability arguments of the previous Chapters, which showed that the localizable Hamiltonians were those given by the Cartan generators of an isometry group G. Thus the path integral for the above dynamical system determines a cohomological. topological quantum field theory which depends only on the second cohomology class of the symplectic 2form (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 Ginvariant, 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 NiemiTirkkonen 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 Gequivariant Agenus of the orbit ON as
g* product ON in
is
a
trivial bundle with trivial Gaction 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
halfsum 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 finitedimensional DuistermaatHeckman theorem
to the Fourier transform of the orbit
localization formula
(4.135))
we
on
the
righthand
side of
(5.27) (i.e.
the
arrive at the famous HarishChandra formula
[21, 67, 140]. The
resulting
character formula associated with the HarishChandra for
mula for the Fourier transform of the orbit is the classical
Weyl
character
N(Hc) lHc W(Hc) [2][4], [51, 87, 117, 143, 156, 1621. Weyl group of Hc, where N(Hc) is the normalizer subgroup of Hc, i.e. the subgroup of g E G with hgHC gHC, Vh E HC, so that N(HC) is the left of HC on the orbit MG action of the fixed of GlHc. points subgroup Let
formula of G
=
be the
=
=
136
Equivariant Localization
5.
Given
w
=
nHC
W(Hc),
E
with
Simply Connected Phase Spaces
on
n
=
the respective adjoint representation formula can then be written as
tr.x
iN
E
N(Hc),
e x(_)
WEW(Hc)
a>O
let
X(')
=
n'Nn be
n1 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 semiclassical 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 semisimple Lie groups is essential to the quantization of spin systems. One important feature of the above topological field theories is that there is a onetoone correspondence between the points on the orbits GIHC and the related this derivation of the
socalled 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 NiemiTirkkonen localization formula (4.130) and which leads directly to the Kirillov character formula (5.27). As
Agenus 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 simplyconnected, 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 simplyconnected
space, the Lie group T(M, g) is locally compact in the compactopen 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 thebasic facts concerning simplyconnected 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 finitelymany linearly independent Killing vectors in IC(M, g). There are 2 important classes of metric spaces (M, g) characterized by their possible isometries. We say that a metric space (M, g) is homogeneous if there exists infinitesimal isometries V that carry any given point x E M to any other point in its immediate neighbourhood. (M,g) is
when M has dimension
138
Equivaxiant Localization
5.
on
Simply Connected Phase Spaces
said to be
isotropic about a point X E M if there exists infinitesimal isomepoint x fixed, and, in particular, if (M, g) is isotropic about all of its points then we say that it is isotropic. The homogeneity con
tries V that leave the
dition
that the metric g must admit Killing vectors that at any given on all possible values (i.e. any point on M is geometrically like any other point). The isotropy condition means that an isotropic point x0 of M is always a fixed point of an T(M, g)action on M, V(xo) 0 for means
point of M take
=
V E
some
IC(M, g),
subject only
but whose first derivatives take
to the
on
all
possible values,
0. Killing equation Lvg dim M homogeneous metric space always admits d linearly independent Killing vectors (intuitively generating translations in the d directions), and a space that is isotropic about some point admits d(d 1) /2 Killing vector fields (intuitively generating rigid rotations about that point). The connection between isotropy and homogeneity of a metric space lies in the fact that any metric space which is isotropic is also homogeneous. The spaces which have the maximal number d(d + 1)/2 of linearly independent Killing vectors enjoy some very special properties, as we shall soon see. We shall refer to such spaces as maximally symmetric spaces. The above dimension counting shows that a homogeneous metric space which is isotropic about some point is maximally symmetric, and, in particular, any isotropic space is maximally symmetric. The converse is also true, i.e. a maximally symmetric space is homogeneous and isotropic. In these cases, there is only one orbit under the T(M, g)action on M, i.e. M can be represented as the orbit M I(M, g) x of any element x E M, and the space of orbits M/T(M, g) consists of only a single point. In this case we say that the group I(M, g) acts transitively on
It follows that
=
a
=

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


+
g, h E
G, which is transitive and for which H is the stabalizer of the point words, homogeneous spaces are essentially coadjoint orbits of Lie groups [68], with H GA the stabalizer group of a point A E g* under the coadjoint action OA Ad*(G)A C g* of G on g*. A sufficient
7r(l). GIH
In other
condition for the coset space GIH fgH : g E GI to be a symmetric space a reductive decomposition, i.e. an orthogonal decomposition =
is that g admit g
=
h (D
hL such that [h
introduce Khhler structures
1 ,
hL]
(the
C
h
[68]. Furthermore,
it is
possible
to
KirillovKostant 2form introduced in the previous Section) on the group orbits for which G is the associated isometry group. These spaces therefore generalize the maximal coadjoint orbit models of the last Section where H was taken as the Cartan subgroup HC and which
Isometry Groups of Simply Connected Riemannian Spaces
5.2
maximally symmetric [68]. We shall
in fact be shown to be
can
examples
later
We shall
see
139
explicit
on.
now
manifold if and
maximally symmetric spaces. uniquely characterized by a special curva(M, g) is a maximally symmetric Riemannian
describe the rich features of
It turns out that these spaces ture constant K. Specifically,
only
curvature tensor of g
are
if there exists can
R,\p,,
a
be written
=
9AMRp1,,,
constant K E R such that the Riemann
locally
=
In dimension d > 3, Schur's lemma
everywhere
almost
K(g,pg,\,

as
(5.31)
gpg,\,)
states that the existence of such
[68]
a
the constancy of K. For d 2, however, this is not the case, and indeed dimension counting shows that the curvature of a Riemann surface always takes the form (5.31). In form for the curvature tensor
automatically implies

=
this
(M, g)
K is called the Gaussian curvature of
case
implies
not constant. The above result
and it is in
general
that the Gaussian curvature K of
maximally symmetric simply connected
a
Riemann surface is constant.
amazing result here is the isometric correspondence between maximally symmetric spaces. Any 2 maximally symmetric spaces (M 1, gi) and The
(M2) 92)
of the
same
dimension and with the
same
curvature constant K
are
M2 between the 2 isometric, i.e. there exists a diffeomorphism f : MI Thus manifolds relating their metrics by gi (x) given any maxi92 (f (x)). onto it isometrically any other one with mally symmetric space we can map +
=
the
same
curvature tensor
(5.31).
We
can
therefore model
maximally
sym
metric spaces by some "standard" spaces, which we now proceed to describe. Consider a flat (d + l)dimensional space with coordinates W', z) and metric I
?ld+ 1
dx,,
=
JKJ
(9 dxP +
1
K
(5.32)
dz (9 dz
realvalued constant. A ddimensional 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 dspace
R
d
space
on
the
dsphere Sd
'Hd.,of constant negative
with its usual flat metric 77Ed

140
5.
Equivariant Localization
on
Simply Connected
Phase
Spaces
(5.33) and the manifest invariances of the it is straightforward to show that the above (5.32) space all admit a + d(d 1)/2parameter 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
ddimensions,
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
realvalued 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 ddimensions, i.e. the semidirect 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 onepoint
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, dHdimensional maximally symmetric subspace Mo (e.g. a homogeneous but nonisotropic 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 ucoordinates 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 vcoordinates alone, and ij (u) is a function of only the ucoordinates that is itself the metric of Mo. As (MO, j)
Isometry Groups of Simply Connected Riemannian Spaces
5.2
141
a dHdimensional maximally symmetric space, it is isometric to one of the 3 standard spaces in dHdimensions 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 ddimensional
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 1dimensional. Consider a 1parameter 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 nontrivial Jacobian
det
For /.t
=
2,..., d

d1 .
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 simpleconnectivity 0 or K > 0 the phase space assumption of this Chapter, that when K M can be either compact or noncompact, but when K < 0 it is necessarily Thus
a
=
noncompact. 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
1parameter
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
noncharacteristic 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
'cigarshaped' geometries
that
are
described in
typical black hole theories
[164]. The
only other
case
left to consider here is when
dimensional isometry group. In this
case we
have 2
(M, g)
independent
has
a
2
vector fields
Isometry Groups of Simply Connected Riemannian Spaces
5.2
145
obey the'Lie algebra (2.47) with possibilities 1, algebra either the isomea, b, it is or is fabc 0, abelian, nonabelian, 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 nonabelian 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
=
eX2
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 realvalued 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
2dimensional
phase

space is either
+ y
(5.61)
straightforward
to
(5.62)
ay), again contradicting
the
maximally symmetric with
a
3dimensional isometry group, or it admits a Iparameter group of isometries (or, equivalently, has a single 1dimensional maximally symmetric subspace), because the above arguments show that it clearly cannot have a 2dimensional isometry group. The fact that there are only 2 distinct classes of isometries in 2 dimensions is another very appealing feature of these cases for the analysis which follows. For the remainder of this Chapter we shall analyse the
146
Equivaxiant Localization
5.
Simply Connected
on
Phase
Spaces
equivariant Hamiltonian systems which can be studied on the various isometric types of spaces discussed in this Section and discuss the features of the integrable quantum models that arise from the localization formalism. We shall primarily develop these systems in 2 dimensions, and present higherdimensional examples in Sections 5.7 and 5.8. This will provide a large set of explicit examples of the formalism developed thus far and at the same time
clarify
some
other issues that arise within the formalism of
path integral
quantization.
5.3 Euclidean Phase
and
Spaces Holornorphic Quantization
begin
We
study of general localizable Hamiltonian systems with the case 0. The conformal factor phase space M is locally flat, i.e. K (5.49) and (5.52) then satisfies the 2dimensional 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
=
5z 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 2dimensional
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:
2form
(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)
+
afvf
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 Ginvariant 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 2form which is invariant under the full Euclidean The fact that the
symplectic
2form here is
volume form associated with the



148
5.
Equivariant Localization
on
Simply Connected
Phase
Spaces
group, i.e. invariant under rotations and translations in the plane, and it is the Kdhler form associated with the Kdhler metric (5.49) and
The form
on
M
(5.64).
(5.71)
planar equivariant Hamiltonian systems illustrates how the integrable dynamical systems which obey the localization criteria depend on the phase space geometry which needs to be introduced in this formalism. These systems are all, however, holomorphic: copies of the same initial dynamical system on R 2 defined by the Darboux Hamiltonian for the
HOD( z,. )
=
Qz.
+ dz + a. +
CO
;
(5.72)
C
E
z
or identifying z,. iq with (p, q) canonical momentum p variables, these dynamical Hamiltonians are of the form =
D
H0
Thus the in the
(p, q)
dependence
on
=
p(P
2
+ q
phase
the
2)
+ CelP + C92q +
and position
(5.73)
CO
space Riemannian
geometry is trivial
that these systems all lift to families of
holomorphic copies of the planar dynamical systems (5.72). This sort of trivial dependence is to be expected since the (classical or quantum) dynamical problem is initially independent of any Riemannian geometry of the phase space. It is also anticipated from the general topological field theory arguments that we presented earlier. Nonetheless, the general functions Ho(z,, ) in (5.71) illustrate how the geometry required for equivariant localization is determined by the different dynamical systems, and vice versa, i.e. the geometries that make these dynamical systems integrable. This probes into what one may consider to be the geometry of the classical or quantum dynamical system, and it illustrates the strong interplay between the Hamiltonian and Riemannian symmetries that are responsible for localization. Thus essentially the only equivariant Hamiltonian system on a planar symplectic manifold is the displaced harmonic oscillator Hamiltonian sense
HOD and in this
=
Q(z
case
action with the
+
a)(.
+
a)
=
Q
J(p + a, )2 + (q + a2 )21
+
(5.74)
CO
we can replace the requirement that H generate a circle requirement that it generate a semibounded group action.
To compare the localization formulas with
ementary quantum mechanics,
we
some
wellknown results from el
note that the Hamiltonian
(5.73)
can
only
describe 2 distinct 1dimensional 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 NiemiTirkkonen 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 NiemiTirkkonen localization formula r eiO with r C R+ polar coordinates z
for the harmonic oscillator. In
and 0 E
[0, 2r],
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(Tt)/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 Tperiodic 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
27rn any initial condition z(O) E C leads space of critical trajectories is non
Tperiodic 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 Tperiodic 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 theAgenus (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' T02iTOop/2v 2_ 0
,
f dp dq 00
e'TP2/2 (5.78)
150
5.
Equivariant Localization
Simply Connected
on
Phase
Spaces
which also coincides with the exact propagator trII eiTp2 /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 HeisenbergWeyl 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
(infinitedimensional)
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)
=_
HeisenbergWeyl Zn
e'at 10) n=0
These states
are
normalized
=
Iz)t,
and
(581)
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
er
2
n+m
E Vn!m! f dO ei(nm)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
HeisenbergWeyl son
=
IZI *"D
=
(5.88)
1
algebra representation of the HeisenbergWeyl 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.
2form 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 1form on
R
2
can
Localization
(5.90)
on
and the
=
dz (9 d,
11 d1z)) 11
=
globallydefined
2
tr
eM
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))
27r
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 HeisenbergWeyl 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 DuistermaatHeckman 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
Wittentype 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 2invariant
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 HeisenbergWeyl 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 semisimple
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 nonhomogeneous phase spaces and even nonsymmetric 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 semiclassical localizations for quantum systems. We shall also see that they are related to the geometric quantization of dynamical systems
[1721. Given any irreducible unitary representation D(G) of the group G and normalized state 10) in the representation space, we define the (normal
some
ized)
state
1g) by 1g)
=
D(g)10)
(5.96)
154
5.
Equivariant Localization
on
Simply Connected Phase Spaces
Dg denotes Haar measure of G, then Schur's lemma [162] and the pleteness of the representation D(G) implies the completeness relation If
dimD(G) vol(G)
f Dg Ig)(gl
=
com
(5.97)
1
G
Following
the derivation of Section 4.1, it follows that the
associated with can
be
operator 7
an
acting
on
partition function
the representation space of
D(G)
represented by the path integral
f Dg (gl eM Ijg)
trD(G) eiT) 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 Hcbundle 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 semisimple Lie groups as sections of a line bundle L 4MG associated to the principal fiber bundle G + GIHC is known as the BorelWeilBott 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

W11V1
compactified plane CU fool.
(5.117)
(5.118),
the metric tensor transformation law and
algebra
that the coordinate transformation above must
( aw,
1
(1
+
WliV1)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)
4K1f (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 BorelWeilBott 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 zaxis. Thus in this
=
case
S2 is actually
162
Equivariant Localization
5.
Simply Connected Phase Spaces
on
naturally the configuration space for a spin system [156], which has on it a sYmplectic structure and so the corresponding path integral can be regarded as one for the Lagrangian formulation of the theory, rather than the Hamiltonian one [139, 156]. This is also immediate from noting that the stereographic complex coordinates above can be written as natural
z
in terms of the usual
J3
in
(5.131)
to
an
additive
=
e O tan(0/2)
spherical polar coordinates (0, 0), height function (2.1)
so
that the observable
of S2 with
coincides with the
a
=
I
(up
constant),
the Khhler geometry above becomes the standard and the kinetic term in the action is
round geometry of S2 ,
01, (X).t" after
(5.135)
=
Cos
O
(5.136)
integration by parts over t. The classical partition function, evaluated 2.1, yields the usual Langevin formula for the classical statistical mechanics of a spin system. Alternatively, the motion of the precessing spin can be reduced to that of a charged particle around a monopole which is isomorphic to the problem for the motion of an excited diatomic molecule where the electrons are in a state with angular momentum j about the axis joining the nuclei (i.e. a rigid rotator with fixed angular momentum j about an
in Section
axis) [156].
on the particle, dO of the monopole located at the magnetic field w center of the sphere, and the potential force on the moment, due to the real magnetic field, that leads to the characteristic Larmor precession about the
its
It is the balance between the the Lorentz force
due to the fictitious
=
direction of the field. To construct
Section
j
=
where .
.
we
topological
.
,
Hamiltonian
along the lines
of the
theory of
an irreducible spinj 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)
=
e3P
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) Xkynk
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)
spinj representation of SU(2) and
is
2j+1
djL(j) (z,. ) i
27r
(1
+
zf) 2
(5.143)
dz A J
symplectic 2form 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 onetoone
(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
spinj representation of SU(2)
be derived from the index theorems of the last Section
(see (5.108)).
can
also
164
Equivariant Localization
5.
on
Simply Connected Phase Spaces
where N1
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 1forms (5.91). Similarly, the Khhler structure (5.122) can be represented in the standard and
coherent state form
(5.92).
The WessZuminoWitten quantization condition (4.161) applied to WW implies that j must be a halfinteger, 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 semiclassical 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 1forms 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 BRSTexact action, it still maintains the Schwarztype 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 Schwarztype topological field theory, as opposed to a Wittentype 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
ei(Tt)
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 saddlepoint 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 eiTJ3 I ZO)
a
+
Z04)j (1
+
+
(5.151)
Z2, 2)j
direct calculation is
1
(I
eijT
+ 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
nonperiodic boundary conditions
of the binomial theorem
application
and integrating
over z
E
C
us
find the partition function
(4.115)
is
discussed above.
regulated using the generic
166
5.
Equivariant Localization
f dy(i)(z,,
ZSU(2) (T)
on
Simply Connected
Phase
Spaces
) ((zl eiTj3lZ))
00
dr
(2j
r
1)(1
+
+
r
2eiT)2j
e
ijT
sin 
(1
0
+
r2)2j
sin
i
eiTm M=j
e
1

2
(5.153)
which also coincides with the exact result
trj eiTJ3
21 (2i + 1))
iTj
e'Ti +
eiTi
eiTi
(5.154)
The
righthand 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 spinj representation of SU(2). Within the framework of the DuistermaatHeckman 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 nonvanishing
version
components
G'Vw)'Z and
consequently
the Dirac
=
iJO) (Z" O/i 3
(1LV(i))2
Agenus
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 27rn,
n E Z, the only Tperiodic 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 nondegenerate. How27rn, n E Z, we find Tperiodic 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 2form w(j) on the sphere [85], i.e. the integervalued 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 NiemiTirkkonen 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 2form has the
nonvanishing 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 Agenus 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
1Y I+y
exp
(T (1y)) f
1+y
[ijT (I Y)l
the ParisiSourlas integration formula
(5.162)
1 + Y
[136]
00
1 7r
1
d 2x
d77 d
L(X2
+
7A
R20AIR2 we
obtain from
du
=
du
L(oo)

L(O)
(5.163)
0
(5.161)
the partition function
ZSU(2) (T)

sin(Ti) / sin(T/2)
(5.164)
168
5.
Equivariant Localization
Introducing
the
Weyl shift j
Phase
Spaces
in + .1 2
(5.164) then yields the correct Weyl (5.153) for SU(2) 5 Note that (5.163) shows explicitly localization in (5.161) comes directly from the extrema of the height j

character formula how the
function at
As
Simply Connected
on
z
=
oo
.
and
z
0.
=
final application for the above dynamical system, we examine the quadratic localization formula (4.150). Now the (degenerate) Hamiltonian is a
.F(j(j)) 3
Following written
the
same
steps
as

(j(j))2 3
above,
(1+ Zf) 1
j2
=

Z'
(5.165)
the localization formula
(4.150)
can
be
as 00
W
ZS U (2) (T I (J3' ) )
f doo f 00
i
2

v
47riT
dz d2
dq d L(Oo, z2 +,q )
R20A1R2
00
(5.166)
where T00 1Y 2 1+y
L(00, y) sin
and
we
formula
010 (10) 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 ParisiSourlas 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 Gindex 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 noncompact manifold, and we can map it onto the maximally symmetric space V, the Lobaschevsky plane (or pseudosphere) 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 sym2form 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 pseudospherical coordinates pseudosphere, 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 pseudosphere itself is represented by the interior of the disk. The explicit transformation in terms of pseudospherical 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) upperhalf plane
is
g, 12
The
path integral
over
such
studies of quantum chaos

=
IM(6)2d6 (9 d
hyperbolic geometries
[311.
(5.172) arises in
string theory and
170
5.
Equivariant Localization
The most
general
on
Simply Connected
localizable Hamiltonian in
Phase
Spaces
hyperbolic phase
a
space
geometry is therefore
(
S? H_ (z,
2)
=
IKI 4
f
+
(Z)1(2))
al(2)
+
IKI 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 quasidisplaced reducing Darboux phase space is now noncompact, we can again weaken the requirement of a global circle action on the phase space to a semibounded 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 2form 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 noncompact 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 infinitedimensional
for the discrete irreducible
1,
[137]. 32 , 2, 55,... 2
The
=
because of the noncompactness 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
eiT(k+n)
iTS3 n=O
5.7 Localization of Generalized
=
2i
e
iT(ksin
T
2
(5.184)
2
Spin Models
and Hamiltonian Reduction The
explicit examples
have given thus far of the localization formalism quantum cases have, for simplicity, focused on dy
we
in both the classical and
namical systems with 2dimensional phase spaces. Our main examples have been the harmonic oscillator, where the localization is trivial because the Hamiltonian is a quadratic function, and the spin partition function, where
172
Equivariant Localization
5.
Simply Connected Phase Spaces
on
the exactness of the
stationaryphase 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
higherdimensional 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 ItzyksonZuber 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

U1 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 leftright 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
ItzyksonZuber 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 DuistermaatHeckman 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 ItzyksonZuber integral (5.185), we
follows from
sets
(5.196) (for
=
a
=
0 in Section
2.1).
5.7 Localization of Generalized
There
which a
are
unitary
Spin Models and Hamiltonian Reduction
various extensions of the above
are
relevant in matrix model theories matrix
generalized
[92].
classical
First of
all,
175
spin model
we can
consider
integral of the form
I[A; T]
=
N
f
DU exp
iT
1:
Aij I Ujj 12
(5.200)
i,j=l
U(N)
The stationary conditions for the Hamiltonian in
(5.200)
are
N
EUjj(AjjAjk)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
saddlepoint
(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 higherorder corrections in (5.202) to the lowestorder 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 ItzyksonZuber model (5.185),
for the correlation functions in the I [X,
Y; T]
kj1j***kIII
=
f
DU
Uj1j1
...
U(N) x
which
U iPjP
(Ut)ki I,
...
(Ut)kplp (5.204)
eiT tr(UXUtY)
only nonvanishing correlators because of the U(1) phase ine"9Uij of the integral (5.185). The evaluation of the unitary matrix integrals (5.204) is very important for matrix models of induced gauge theories and string theories [92] and has been a difficult problem that has are
variance
the
Ujj
+
received much attention in the last few years. For instance, using GelfandTseytlin coordinates on the group manifold, Shatashvili [1541 has shown that
176
Equivariant Localization
5.
some
(5.204)
of the correlators
mulas, such
Simply Connected Phase Spaces
on
explicitly given by
are
complicated for
very
as
I[X,Y;T]'IJ1JJ2, "" k2,1...k,,l IJP *
Jil ki Ji2 k2
*
*
00
f! riN1 R=1
Jjp kp
(iT)(N2)(N1)
6 [X1,1A (Y2 i
...
7
f
YN,
_cO
(k=1E N
X
iTy,
exp
NI
Xk
E

k=1
Ak)
N1
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 2planes (see below). In the next Section we shall see how to explicitly relate the space CpN to a (nonmaximal) 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, 27r),
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 DuistermaatHeckman 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 Aplane in (CpN is the conserved charge
2JO It4 1
and
an
integrable
functions
(5.212),
Hamiltonian
can
on
For
the
(5.212)
+EN V=1
be constructed
0. More
e'6" A.
as
the
sum
of the N
height
178
5.
Equivaxiant Localization
on
Simply Connected
Phase
Spaces
N
(5.213)
HN
The conserved charges (5.212) are the action variables of the above integrable model and the associated angle variables OA c [0, 27r] are the usual polar angles of the Aplanes (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
,
Nl
=
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) eioN'
sin(ON 1/2) sin(ON /2) eioN (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 2form (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 
5pA
E /I=
A
aolt
(5.216)
integrable system on the phase space (CpN is therefore once again isomorphic to a linear combination of independent harmonic oscillators as above, generalizing the dynamics of our standard spin example. The relation between such generalized spin systems and systems of harmonic oscillators, and hence the exactness of the localization formulas, can be understood from a slightly different perspective than the usual localization symmetries provided by the underlying equivariant cohomological or supersymmetric structures of the dynamical system. To see this, we evaluate the classical partition function explicitly for the above dynamical system by extending the integration over (CpN to the whole of (CN+l via a Lagrange multiplier which enforces the constraint (5.206). This leads to This
5.7 Localization of Generalized
ZN (T)
Spin Models and Hamiltonian Reduction
2J dtf' d ' IIN p=j N(j + EN qrl)N+l
eiTHN( , ;O)

Ir
S2N+IISI
,,=,
00
f
dA
N+1
j
27r
d2F1 &A
ri
R2N+2
00
(5.217)
7r
tj=1
1)
N+1 x
where the Gaussian
Tl.(N+ T "D
exp
179
1)
(Z
1
7
0)
E
+ j'X
angular parameters in (5.217) are related by OA integral over CN+1 in (5.217) can be performed 00
ZN (T)
f
=
01'
=
to
_
ON+1 The .
yield
N+1
dA
%+
e'A
TOA
(5.218)
A
00
remaining integration in (5.218) can be carried out by continuing the integration over a large contour in the complex plane and using the residue theorem to pick up the N + 1 simple poles of the integrand at A TOA, each of which have residue 1. Thus the classical partition function is The
=
N+1
ZN (T)
=
E
e
iT01'
j1=1
It is
readily verified
that
(5.219)
11 V014
i
T(Ov

(5.219)
P)
coincides the DuistermaatHeckman integra(5.213) which has precisely N + 1 stationary
tion formula for the Hamiltonian
points [52]. (5.219) therefore represents the TN equivariant cohomology of the manifold CpN. The
explicit evaluation above illustrates
an
interesting feature of the lo
calization in these cases, as was first pointed out in [44, 52]. This is seen in (5.217) the dynamical system describing the SU(N + 1) isospin is the 
Hamiltonian reduction of
quadratic
Hamiltonian
on
a
larger dynamical system which is described by a 2N+2 and for symplectic manifold R
the Darboux
which the localization is therefore trivial. The reduction constraint function
(5.206)
commutes with the Darboux oscillator Hamiltonian
P(z,, ), HD( +')I N
so
that the constraint function
P(z,, )
=
(,+ )
0
(5.210),
i.e.
(5.220)
WD
determines
a
first class constraint
on
dynamical system and is therefore a symmetry (or conserved of the classical dynamics. This commutativity property is in fact the
the Darboux
charge)
correspondence between the integrations in (5.217), CN+1 is restricted to the symplectic subspace dedynamics termined by the constant values of the conserved charges P(z,, ) (according crucial mechanism for the as
then the
on
180
5.
Equivariant Localization
[7]
to the reduction theorem

on
see
Simply Connected Phase Spaces
(4.144)).
reduction mechanism here is the action of the usual matrixvector
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 DuistermaatHeckman 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 CalliasBott index theorems from dimensional Section
4.2).
AtiyahSinger
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 DuistermaatHeckman 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 semiclassical 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 timedependent 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
(timeindependent) vectors
be written
can
as
R(t) with
g(t)
Ad*(A')g(t)
=
=
g(t)A'g(t)'
(5.221)
timedependent 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
S2N1
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
nonnegative
a
representation,
way that the maximum stabalizer
that the
loop space path integral will be Ad* (SU(N + 1))A' SU(N + OA1 appropriate to the geometry of the given
=

coadjoint orbit (e.g. the normalization of the coherent Kdhler potentials of the orbits in the manner described
states
generates the
in Section
5.4).
The
182
5.
Equivariant Localization
existence of such
weight
a
state is in
Simply Connected
Phase
Spaces
general always guaranteed by the Borel
[68, 142].
WeilBott 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 GellMann basis have the (diagonal) matrix elements
posed
[H,,,]ij
into the
sum
1 =
V2m(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 eM 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
[dlP) (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 multidimensional generalization of the Kdhler polarization for the spin propagator in Section 5.5 above. The classical equations of motion are is the
for the coherent state
+
2U ' '
whose solutions in the
periodic
=
2UP '
0
stereographic coordinates (5.214)
=
are
0
the
(5.234) conditionally
motions
0'(t)
=
0'(0)
0'(t)
generalizing the results of Section In any case,
1CM
we
=
0'(0)
+
2Jdt
(5.235)
5.5.
arrive at the propagator
+
T)
=
EN
1 1 '=
'Q
1
2iJi"T
)2J

+
rk E,, 11 14) (1 + E,, 2 t2 ) J J
e
2iJCoT
(5.236)
184
Equivariant Localization
5.
on
Simply Connected
Phase
Spaces
and the associated quantum partition function
f dy(j)(C,t) (( j e_i? TI6))
ZSU(N+1)(T)
N+1
E
e2iJO'T
a=1
11 '301a
(5.237)
1
I

e2iJ(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
HeisenbergWeyl 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
multidimensional
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
(5242)
using the important property
I)q (D), f J] it is
of
straightforward
to show that
(5243)
=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 ZaLa)

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 27r
I
J1
e2iJA

2iJ61T+iA e
0
J
d aw 21r
W2J+N
we
integral
have transformed the
up N + I
simple poles each
w

e2iJj11T
angular integration
S1. Carrying
the unit circle
over
J1 a=1
S1
where
(5.245)
N+1
in
(5245)
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 noncompact hyperbolic NJ space D (the complex open (N + l)dimensional Poincar6 ball [68]) with the associated SU(N, 1) coherent states, generalizing the results of Section tion is
systems
186
5.
5.6 for
Weyl
Equivariant Localization
SU(1, 1) [52].
In
on
Simply Connected
Spaces
particular, the localization formulas all lead
to the
SU(N, 1)
character formula for
N
ZSU(N,I)(T)
=
1
]1
eiCoKT
I
Oe=1
The
Phase
CpN model above
(5.246)
eij"T

special case of the more general Kdhler space Geometrically, this is defined as the k (N k) complexdimensional space Gr(N, k) of kplanes through the origin of (CN (note that Gr(N, 1) CpN1). 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 KirillovKostant 2form
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 noncompact 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 noncompact 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 ItzyksonZuber 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
DuistermaatHeckman formula Gr(N,k)
Z.'I
2,7ri
(T)
k(Nk),
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
Xnt1 Xnt, be parametrized
=
can
(5.258)
=
by
=
Zil Zj The canonical Iform
=
on
det[Zl,..., ZNI
6ij
=
(5.259)
1
the orbit is then NI
0Inij where Ji
=
x, +...
constraint in
tr
(Xgldg)
=
JiZtdZ.. i
i
(5.260)
+XN1 ! 0 and we have used the determinant The associated Kdhler 2form is
+2xi
(5.259).
N1
.1nij
and the
=
dO {nil
isospin element (5.258)
can
N1
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 (5262) 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 PoissonLie 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 noncommutative 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 semiclassical 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 ei02 (Zl) Z2) Z3) ( Z3), respectively [66, 86] (so that H generates Z1, Z2, the symplectic action of the 2torus 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 eid 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 Ginvariant metric
190
5.
Equivariant Localization g
Simply Connected Phase Spaces
on
tr(F
=
Ff)
0
(5.265)
hldh is the adjoint repredefined on the group manifold of G, where sentation of the CartanMaurer 1form which takes values in the Lie algebra
geodesic motion on G, the Hamiltonian operator Schr6dinger polarization is given by the LaplaceBeltrami 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 semiclassical approximation is given by a sum over the lattice Z' in rdimensional Euclidean root space generated by the simple roots of g. The timeevolution kernel is therefore a
where h E
function of the rdimensional vector
formulas for it
IC (a;
T)
can
(
be shown to
27riT
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=(kl,...,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) eic(,\)T
(5.269)
p?)
(5.270)
AEZ'
where
C(A)
+
p,)2
_
Quantization
5.9
on
NonHomogeneous
Phase
Spaces
191
eigenvalues of the quadratic Casimir operator E,,(Xa)2 in the reprehighest weight vector A. Similar considerations also apply to nspheres 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 instantonantiinstanton pair with 2 chiral spins [100]. Notice that these localizations also apply to the basic integrable models which are wellknown to be equivalent to the group geodesic motion problems above, such as 2dimensional YangMills theory, supersymmetric quantum mechanics and CalegoroMoser 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.
NonHomogeneous Phase Spaces
Thus far in this
Chapter we have examined the localizable dynamical systems phase spaces which are maximally symmetric and those with multidimensional maximallysymmetric 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
nonconstant 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 nontrivial 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 nonabelian. 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 Iparameter 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 nonconstant function
on
M,
=
so
1, V/2
but
we
that in these
=
0. For now,
shall
soon see
192
5.
Equivariant Localization
Simply Connected
on
Phase
Spaces
how,
once a given isometry of the dynamical system is identified, it can be fixed to suit the given problem. For a Hamiltonian system (M, W, H) which generates the flows of the given isometry in the usual way via Hamilton's
equations,
the
defining
(5.55)
condition
for the coordinate function X 2
now
reads
JH, X21,0
=
LVX2
=
(5.271)
0
which is assumed to hold away from the critical point set of H (i.e. the zeroes of V) almost everywhere on M. This means that X2 is a conserved charge of
the given dynamical system, i.e. a GMinvariant 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 2dimensions this
X
and
so even
in the
2
(5.272)
nonhomogeneous
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 2form now only implies that, in the new xlcoordinates, 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 2form 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 2form 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
fewparameter 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 2form on M R and hope to has been
=
Quantization
5.9
NonHomogeneous
on
Phase
Spaces
193
obtain localizable
examples of 1dimensional 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 (5272). The Hamiltonian vector field in
that the function
F(H)
vr2_H_
=
coordinates has the
in
single nonvanishing 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
9009rVo
=
0
(5.276)
0
The 3 equations in (5.276) can be solved in succession and the general solution has the form
by integrating
them
0
goo
f (r)
gro
f (r) VO
I 00
dO' ar
( Vo,I)+
h
Vo
grr
f (r)
+k(r)
(5.277) f (r), h(r) and k(r) are arbitrary C'functions that are independent of the angular coordinate 0. Note that, as expected, there is no unique solution for the conformal factor W in (5.49), only the requirement that it be radially symmetric (i.e. independent of 0). However, the equations (5.277) impose a much stronger
where
194
5.
Equivariant Localization
Simply Connected
on
Phase
Spaces
requirement, this time on the actual coordinate transformation (5.274). If impose the required singlevaluedness 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 V0
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 1dimensional quantum mechanical Hamiltonians with static potentials which
are
bounded
satisfying the
below, due
to the nonexistence of
Lie derivative constraint in this
a
singlevalued
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 SIaction 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 NiemiTirkkonen formula (4.130), are explicitly metric dependent through, e.g. theAgenus terms, although not manifestly so. Explicitly, the nonvanishing 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 nonvanishing 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)
=
NonHomogeneous
on
Phase
Spaces
e '(')(Qv)o,/2r
195
(5.282)
Substituting the above quantities into the NiemiTirkkonen 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
eiTr 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 rO
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
rO
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 AtiyahSinger index which appears as the NiemiTirkkonen formula for the quantum mechanical path integral is an integral over charac=
=
=
teristic
classes,
it is
manifestly invariant
under CI deformations of the metric
M. The transformation to polar coordinates is a diffeomorphism only on 2 the punctured plane R 101, which destroys the manifest topological invarion
_
of the partition function (at r 0 anyway). For A(O) 4 0, the metric a conical geometry [102] for which the parameter A(O) represents the tip angle of the cone. This example shows that for the localization ance
=
tensor describes
to work the choice of metric tensor is not
completely arbitrary,
since it has to
respect the topology of the phase space on which the problem is defined. As discussed in [40] and [159], this appears to be a general feature of the generalized localization
formalisms,
in that
they detect explicitly the topology
of the
196
Equivariant Localization
5.
on
Simply Connected Phase Spaces
phase
space and this can be used to eliminate some of the arbitrariness of the metric (5.49). Indeed, in the set of prefered coordinates for V it has no zeroes
and
so the critical points are "absorbed" into the symplectic 2form 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, nonhomogeneous 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 welldefined orbits on the Ocircle, 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 NiemiTirkkonen 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 1dimensional
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 3dimensional 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 NiemiTirkkonen 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 nonvanishing component
(5.288) vector
5.9
Quantization
on
NonHomogeneous
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 noncompact range of the hyperbolic coordinate r in the case at hand, we do not encounter a singlevaluedness 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 welldefined function Now the
gr'r
_
12f (r)
=
sinh r
V'r
In the context of
cosh
our
2
coordinates
are
X1
1arctan(sinhr)
=
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 welldefined metric which is translation invariant in the variable (5.291). It is also possible to evaluate the NiemiTirkkonen 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 AtiyahSinger index in (4.130) an explicitly metric dependent quantity. With these appropriate geometric restrictions it is enough to argue that the quantum partition function for the Darboux Hamiltonian
(5.286)
has the form
[40]
00
Zh (T)

E n=1
e
iT/2n2
(5.292)
198
5.
Equivariant Localization
which from
(5.287)
we see
1dimensional
Simply Connected Phase Spaces
on
is indeed the exact
spectral propagator for the
hydrogen [95]. example shows that more complicated quantum systems can be studied within the equivariant localization framework on a simply connected phase space, but only for those phase spaces which admit Riemannian geometries which have complicated and unusual symmetries, such as translations in the coordinate (5.291) above. Thus besides having to find a metric tensor appropriate to the geometry and topology of a phase space, there is the further general problem as to whether or not a geometry can in fact possess the required symmetry (e.g. for Hamiltonians associated with bounded potentials, there is no such geometry). It is not expected, of course, that any Hamiltonian will have an exactly solvable path integral, and from the point of view of this Chapter the cases where the Feynman path integral fails to be effectively computable within the framework of equivariant localization will be those cases where a required symmetry of the phase space geometry does not lead to a globally welldefined metric tensor appropriate to the given topology. Nonetheless, the analysis in [40] for the 1dimensional hydrogen atom is a highly nontrivial 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 nonhomogenous phase spaces which fall into the framework of loop space equivariant localization to character formulas for the associated Iparameter isometry groups GM [159]. For this, we need to introduce a formalism for constructing coherent states assoatom
This
ciated with nontransitive group actions on manifolds [89, 159]. We consider (5.49) in the prefered xcoordinates 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 2form 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) GMinvariant
(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. )
nonhomogeneous
(5.295) Kiffiler mani
5.9
Quantization
F( ') (z, 2) w(w) determines
such that
bundle L(w) Let
Np,
Taylor
the
NonHomogeneous Phase Spaces
on
log f (z2)
=
199
(5.296) symplectic
the first Chern class of the usual
line
M.
+
Nw
0
I is nontrivial and the hyperbolic Riemann surface is nonabelian. 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 noncompact) simplyconnected Riemann surfaces the 2sphere S2, the plane C and the Poincare upper halfplane 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 simplyconnected
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 fixedpoint free biholomorphic action of =
Fh jj2
on
?j2 The
is
PSL(2,R)
group of

transformations
=
as
analytic automorphisms
of the upper
halfplane
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 realvalued. Then
PSL(2,R)action
on
transformations in 2dimensions and 7rl (Zl)
was
taken to be the lattice sub
Indeed, it is possible to regard Zh as a 4hgon 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 noncompact complex manifold which is homeomorphic: to (CM3, 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 globallydefined 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 globallydefined on _rh means that they must be singlevalued under windings around the canonical homology cycles 
228
Equivariant Localization
6.
ae,
bt I h
E
1
Hi (_Th; Z),
Multiply Connected
on
equivalently
or
dV1'
dV"
Spaces
that
(6.82)
h
0
=
Phase
bt
at
Using this singlevalued
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 metricdual 1form
g(V, )
the
we
Al _Th'
E
9(V7
dX
+
*d
+
(6.84)
4
where X and are Clfunctions on Zh and Ah is a harmonic solution of the zeromode Laplace equation for 1forms,
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),
ddimensional Riemannian manifold
it encodes the Riemannian geinto the DeRham cohomology. It is defined as the map
ometry directly
*:
which is given
=
1form,
AkM
AdkM
*
(6.86)
locally by I
*ce
=
(d Xf

A1
k)! ...
Vdetg(X)
Adkil
...
ik
a l
gj,,X,
...
ik
W

(x) dxjl
9jdkAdk A
...
A
W
(6.87)
dXjd k
and satisfies
(,)(dI)k on
AkM.
Using
product fm a A *,8 on each vector space AkM. product possible to show that a differential form Ah as harmonic if and only if It defines
an
inner
this inner
above is
(6.88)
it is
dAh
=
d
*
Ah
=
(6.89)
0
and the Hodge decomposition theorem (6.84) (which can be generalized to arbitrary degree differential forms in the general case) implies that the DeRham cohomology groups of M are spanned by a basis of harmonic forms. The Hodge decomposition (6.84) is unique and the components involved there are explicitly given by I X
=
V2Eh
*
d*
g(V,
V2Eh
dg (V,
(6.90)
6.6 Generalization to
Laplacians V2
where the scalar
*d
h
Hyperbolic
(6.90)
d in
*
Riemann Surfaces
229
assumed to have
are
modes removed. The 1form 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
Poinear6Hodge duality theorem [32],
orthonormal basis of harmonic 1forms
f at
f at, &1h
homology
Poincar6dual 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 curlfree, longitudinal and
divergencefree,
transverse
pieces
as
VZhX
+
VZh
We
can now
write the
general
. The harmonic part
x
Ah accounts for the fact that this 1form may sit in cohomology class of H1 (Zh; R).
a
nontrivial 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 halfplane. 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 metricdual 1form 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 d02/27r. 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 2form of _Th The usual equivariance condition 'C Vv CO The harmonic
=
=
,
=
h
now
becomes h
div_,h where
O(x)
d
is the C'function
(6.93) implies
that it is constant
Vl at
(D on
on
h
+
V2,Ot
defined
Zh7 just
as
by w,,,(x) in
(6.93)
0
=
CD(x),E1,
and
(6.19).
Integrating up the Hamiltonian equations we see therefore that the unique equivariant (Darboux) Hamiltonians have the form
H_r h (X)
=
E I (Mae + W2,3e) 1
t=1C.
(6.94)
230
6.
Equivariant Localization
hle
on
Multiply Connected
Phase
Spaces
realvalued constants and Q, C Zh is a simple curve from basepoint to x. The Hamiltonian (6.94) is multivalued because it depends explicitly on the particular representatives at,0i of the DeRham 2h R As before, singlevaluedness of the cohomology classes in H'(Zh; R) timeevolution 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 nontrivial 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 homologicallyinvariant 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 GaussBonnetChern 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)3h3
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)3h3
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 nonsymmetric 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 nontrivial homology classes of the phase space. The coherent states in the finitedimensional Hilbert space also carry a multidimensional representation of the discrete first homology group, and the localizable Hamiltonians on these phases spaces are rather unusual and even restricted than in the
simplyconnected
The invariant
symplectic Z, as in the maximallysymmetric cases, and it is essentially the global topological features of these multiplyconnected phase spaces which leads to these rather more
2forms in these
severe
cases are
cases.
nontrivial elements of H 2(Zh;
restrictions. The coherent state
Z)
=
quantization of these systems shows
that the path
integral describes the coadjoint orbit quantization of an unusual spin system described by the Riemann surface. These spin systems are exactly solvable both from the point of view of path integral quantization on the loop space and of canonical holomorphic quantization in the Schr6dinger polarization. The localizable systems that
one
obtains in these
cases are
rather
6.6 Generalization to
Hyperbolic
Riemann Surfaces
231
trivial in appearence and are associated with abelian isometry groups acting on the phase spaces. However, these quantum theories probe deep geometric and
topological
features of the
phase
spaces, such
as
their
complex algebraic
geometry and their homology. This is in contrast with the topological quanwe found in the simplyconnected cases, where at best topological path integral could only represent the possible nontrivial 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 semisimple 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 SemiClassical
we
shall examine
We return to the
3 and consider
a
Approximation
a different approach to the problem of localgeneral finitedimensional 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 Iloop 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 higherorder 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 semiclassical approximation as well as new geometric criteria for localization which expand the encountered that
are
into the
time
previous isometry conditions.
approach to the DuistermaatHeckman integration formula using the perturbative loopexpansion 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 3dimensional 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 finitedimensional nor compact and the Hamiltonian flows need not be periodic, but the dynamical system admits an infinite sequence of constants of motion which
localizable and
one
[1741.
are
in involution
The standard localization
needs to resort to
The extension to
an
analysis
so
that it should be
analyses therefore do
not
apply
of the sort which will follow here. We
degenerate Hamiltonians
is
fairly straightforward.
In what fol
lows all statements made concerning the structure of the discrete critical Point set
Mv of H will then apply
to the full critical submanifold.
R. J. Szabo: LNPm 63, pp. 233  268, 2000 © SpringerVerlag Berlin Heidelberg 2000
234
Beyond the SemiClassical 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 nontrivial singular contributions to the usual Iloop 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 nontrivial 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
2form which appears in
of as
7.1 Geometrical Characterizations of the 00
VI'det (x) w
Vdet (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 largeT 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 oddorder moments vanish, since these integrands can



odd functions, and the 2kth 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
stationaryphase expansion 21ri
Z(T)
T
)n
a
term of order
into account the
signature
largeT 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 
stationaryphase 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 loopexpansion of this zerodimensional 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 finitedimensional 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 loopexpansion with the hope
In this Section
we
generalization
of this formula to the
case
where H is
a
degenerate
236
of
Beyond the SemiClassical Approximation
7.
vanishing or nonvanishing of the kloop congeometrical and topological features of the experience now with the DuistermaatHeckman 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 higherloop contributions will for the most part be of a different geometrical nature than the situation that prevails in DuistermaatHeckman 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 lowestorder DuistermaatHeckman term Ao(p) above, but the higherorder 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;, ,
(74)
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 DuistermaatHeckman
=
(1loop) 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 onecomponent 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
Al(x')o9' A(x')]
(7.10)
where
A1(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 higherorder derivatives of H and the connection /, noting that only third and higherorder derivatives of g(x; p) when evaluated at x p are nonvanishing. After some algebra, we arrive at For
=
=
A 1 (P)
Ao(p) 2
li(p) 'v
D,(x)7,(x)
+4y,,(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 torsionfree connection r ',\, of the tangent bundle TM.
It is
variant under local
For now, we need not assume that r is the LeviCivita connection associated a Riemannian metric on M. Indeed, as the original dynamical problem
with
is defined
only
symplectic geometry, not a Riemannian geommanifestly covariant in its own right without reference to any geometry that is external to the problem. All that is required is some connection that specifies parallel transport along the fibers in terms of
etry, the expression
(7.12)
a
should be
of the tangent bundle and allows
us
to extend derivatives of
quantities
to
an
neighbourhood, rather than just at a point, in a covariant; way. The Hessian of H can be written in terms of this connection and the
entire
associated covariant derivative
7i(x),,, and, using d r, appearing in (7.12) in
=
as
V,,V,,H(x)
we can
+
rA (x)49,\H(x) Mv
(7.13)
write the third and fourth order derivatives
terms of V and r
by taking derivatives
of
(7.13).
238
7.
Beyond the SemiClassical Approximation
Substituting
these
complicated derivative expressions into (7.12) and using 0 on MV, after a long and quite tedious calculation we arrive at a manifestly covariant and coordinateindependent expression for the 2loop 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
(714) 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
(715)
have introduced the
1form A
,6,, (x) It is
intriguing that the
=

r,, ,\ (x)
=
V,, log hL (X)
covariantization of the
replacing ordinary
volves
y,, (x)
(7.16)
2loop expression simply inones V, noncovariant
derivatives d with covariant
connection terms y with the 1form 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 LeviCivita connection compatible with a metric g, i.e. 0,, log vqetg and the 1form components (7.16) become
Vg
=
0, then P\ JLA
A ,
=
a,, log Vdet(g1 w).
=

The covariantization of the
higherloop terms is hopelessly complicated. if, however, the 2loop, 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 DuistermaatHeckman formula to all orders in the loopexpansion. To see this, we exploit the topological invariance of (7.14) and apply the Morse lemma [111] to the correction terms (7.5). This says that there exists a sufficiently small neighbourhood Up about each critical point p in which the Morse function H looks like a "harmonic oscillator", We note that
H(x) so
=
H(p) _(XI)2 _(X2)2_..._(X,\(p))2+(X,X(p)+1)2+...+(X2n)2
,
Up (7.17)
x
E
that the critical point p is at x 0 in this open set in M We shall call these coordinates', and this result simply means that the symmetric ma=
.
'harmonic trix
H(x)
critical nates
can be diagonalized constantly in point. Given that the quantity (7.5)
(although
not
manifestly),
we can
an
neighbourhood of the independent of coordi
entire
must be
evaluate it in
a
harmonic coordinate
7.1 Geometrical Characterizations of the
system. Then the Gaussian deviation function the the
neighbourhood Up and only series (7.3) is simply 27ri
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
2loop 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)
higherloop terms in these coordinates can be written as derivative on the 2loop contribution A, (x) as prescribed by (7.18), all
operators acting
corrections to the semiclassical
approximation vanish. In the
case
of the
DuistermaatHeckman 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 2loop correction term would not imply the vanishing of all higherorders, 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 WKBexact. 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 2loop 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 loopexpansion above which can be used to classify the localizable or nonlocalizable 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 nondegenerate 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 SemiClassical
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 2dimensional 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
globallydefined
(7.20)
=!Pg
VVH
C'function
on
M, and for which the
connection F used in the covariant derivatives V is the LeviCivita connection
for g. This
means
coupled
the
that, given
!g(x)g,,,(x) rA
MV
consistently
a
Hamiltonian H
system of nonlinear
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 (720) 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 inductivelydefined metric in (7.21) is that it allows a relatively straightforward analysis of the 2loop, correction We shall
see
7.1 Geometrical Characterizations of the
Loop Expansion
terms to the DuistermaatHeckman 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 LeviCivita expressed in terms of the 1form. 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 2form 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 LeviCivita connection associated to some generic, globallydefined 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 nonvanishing components of J are given by (5.9), so that there is "almost" a separation of the tangent bundle into holomorphic and antiholomorphic 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 subcollection of subsets of the differentiable structure determine a local complex structure (but recall that any Riemann surface is a complex manifold). Again in 2dimensions this means that then VW 0 In
2dimensions, 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 2ndimensional 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 SemiClassical
=
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,,wXPo9pH + 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 anticommutator
[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
nonsingular (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 2loop correction term, i.e. the DuistermaatHeckman theorem. Indeed, from the analysis of the last 2 Chapters we have seen that most of the localizable examples fall into these Kiffilertype, 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,\VlIV,
Vd_et;1_tv(x)
=
VV
as
(7.33)
homogeneous Kdhler manifold, when g is integrated to be (7.25), this generalizes that observed for the height function of the sphere in
relation
Section 5.5.
particularly interesting here is that conversely the localization partition function determines this sort of Kdhler structure on M. The Lie derivative (7.29) of the metric (7.21) is easily seen to be zero in a neighbourhood of the critical point. Conversely, if the Lie derivative of the What is
of the
7.1 Geometrical Characterizations of the
Loop Expansion
243
metric in (721) vanishes on M, then it induces a Khhler structure (i.e. VW 0). Recalling from Section 3.3 the proof of the DuistermaatHeckman 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 globallydefined 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
FubiniStudy
*
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
=
FubiniStudy
The existence of
(7.8).
=
as a
metric in the fibers
Kdhler structure
the curvature 2forms of the associated connections Q
holomorphic line bundle symplectic line
the standard
(7.8),
on
05 log hL
(7.34)
holomorphic and antiholomorphic 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 2form
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 upperhalf plane is
contractible. Thus the existence of a Khhler structure for which A is not
a
problem. However, for the Killing equation for
g
=
J

W
must be invariant under the flows of the Hamiltonian vector field
=
to
V,
V log hL 0 hold, J itself =
i.e.
Cv J
=
0.
244
which is a
Beyond the SemiClassical Approximation
7.
a
point p
subbundle 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 2form will depend on where these 2forms 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 globallydefined 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
Clfunction
=
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
nonconstant function
on
M
7.2 Conformal and Geodetic Localization
Symmetries
245
corresponding to nonhomothetic 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 divergencefree, tr /,tv VtVA 0. We shall show that this conformal symmetry requirement on the dynamical system also leads to the DuistermaatHeckman 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 Iloop 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 DuistermaatHeckman 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 divergencefree with
respect
to g
(symmetric)
(see (5.51)). In 2dimensions, 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 Clfunctions
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 SemiClassical
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 (723) 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 generallycovariant expression (7.14) for the firstorder correction to the DuistermaatHeckman 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 reevaluate the function Z(S) in (2.128)
ization
generic (not necessarily equivariant) metricdual 1form,3 ivg of the (i.e. with no assumptions for now about the symmetries of g) and the equivariantlyclosed 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 CartanWeil
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) esDv)3 M
(7.43)
7.2 Conformal and Geodetic Localization
The
larges
limit
integral
(7.43)
in
did in Section 2.6 to arrive at the
S?V
the 2form
=
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 2forms 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 H1wg1LVg12) (p) 00
+
J TZT__)n I ds
0
0
(n

g(V,
A
(iTw
g(V, 1)!
A
(Lvg)(V, )
1)!

s0v)
n1
am
00
(iT)n J
eiTHsKV
ds
s
I
eiTHsKV
(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 DuistermaatHeckman 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 sintegrations 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 semiclassical approximation to the partition function exact. This is in contrast to the loopexpansion we studied earlier, where the corrections to which holds for
an
=
=
the DuistermaatHeckman formula
derivatives.
(7.46)
were
therefore represents
expansion that explicitly takes into
a
not
just
some
combinations of Lie
sort of resummation of the
account the
loopgeometric symmetries that
248
7.
Beyond the SemiClassical Approximation
1loop approximation exact. We shall see soon that it is quite conpredicted from the loopexpansion, 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 sintegral 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
JeffreyKirwanKalkman
connection 1form 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 Hessianmetric ansatz which was discussed in the last Section. In particular, (7.46) establishes that with this Hessianmetric substitution and the appropriate extension away from the critical points of H (the terms proportional to VH in (737)) the corrections to the DuistermaatHeckman integration formula vanish to all orders of the loopexpansion (and not just to 2loop 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 nonvanishing 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 1form 0 it is easy to show from (7.35) that away from the critical point set of the Hamiltonian it satisfies CV,6 0, i.e. '3 is then an equivariant differential form on M MV. Thus away from the subset MV C M the conformal Killing condition can be cast into the same equivariant cohomological framework as the isometry condition by a rescaling of ever, that because of
is
+
=

the metric tensor in
(7.35),
The rescaled metric
G,,,(x)
gttv is
>
Gl,,,
=
only defined
g1,,1g(V, V), on
M

for which
MV, but
we
LvG
=
0.
recall from
Section 2.5 that all that is needed to establish the localization of the
zeroes
of the vector field V
(i.e.
the equivariant localization
(3.52) onto principle) is
(or equivalently an equivariant differential form everywhere on M except possibly in an arbitrarily small neighbourhood of MV. The fact that the weaker conformal symmetry condition is equivalent to the isometry condition in this respect is essentially a an
,3)
invariant metric tensor
which is defined
7.2 Conformal and Geodetic Localization
Symmetries
249
consequence of the fact that the differential form 3 ivg above is still a connection 1form 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 0form component to produce an exponential damping factor, since higherdegree forms will contribute only polynomially in the Taylor expansion of the exponential. This is guaranteed only if '3 has a 1form component. Thus it is only the 1form 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 1form. Furthermore, the 0form part Vi',3,, of DV,3 must attain its global minimum at zero so that the larges limit in (7.45) yields a nonzero 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 1form of V with respect to a globallydefined Euclidean signature Riemannian structure on the phase space M. In addition, for a compact phase space M the conformal group is in general noncompact, 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 DuistermaatHeckman localization leads to globally different sorts of dynamical systems, one would like to construct examples of systems with nontrivial (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 wellunderstood [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 ddimensional
transformations.
subgroup
(e.g.
of socalled
translations of
special
r
conformal
250
7.
Beyond the SemiClassical Approximation
An interesting
example is provided by the flat complex plane C. Here algebra is infinitedimensional 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 CauchyRiemann equations in (5.51) (the other one represents the divergencefree 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 firstorder linear
partial
differential equation
VZ09ZWZ2
+
V29 WZ2
T(Z" )WZ2
=
(7.51)
easily solved by separation of the variables z,; . The solution symplectic 2form 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 )n1 i0i
are
the coefficients of the
partial
(VZ )_ 1=
1
!)_3
aj
(7.54)
decomposition n
1+ E
_z
singlevalued 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 realvalued. The Hamiltonian equations (5.68) can now be integrated To
ensure
that
to all have the to
(7.52)
fraction
ai
is
a
function
.


,
on
7.2 Conformal and Geodetic Localization
up with the vector field we
find the
family
(7.49)
and the
sYmplectic 2form, (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 nondegenerate
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 DuistermaatHeckman
point set,
integration formula, namely Z(T) 27ri,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 2form 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 (noncompact) conformal group may lead to interesting new structures in other instances which usually employ the full isometry group of the phase ple harmonic
space, such
as
more
is not
details about the
the Witten localization formalism of Section 3.8.
Other geometric alternatives to the Lie derivative condition LVg 0 have also been discussed by Mirki and Niemi in [84]. For instance, consider the =
alternative condition
252
Beyond the SemiClassical Approximation
7.
V"V.Xv/A
=
(7.58)
0
to the
VA are Killing equation, which means that the Hamiltonian flows _V, geodetic to g (see Section 2.4). From (7.58) it follows that the Lie derivative =
of the localization
LV,3
=
1form,3
VP(gpXVIV\
+
=
ivg
can
be written
g1,,\VpV\)dx1'
=
as
VPgp,\V,,VXdxl'
div Comparing this with the CartanWeil 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 equivariantlyclosed 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 nontrivial 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 2form 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 S2v (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 DuistermaatHeckman Formula
253
7.3 Corrections to the DuistermaatHeckman Formula: A Geometric Approach The integration formula
(7.46) suggests
a
geometric approach
tion of corrections to the DuistermaatHeckman 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 semiclassical approximation is not exact, are global obstructions to extending these locally invariant metric ten
to globallydefined 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 singlevalued 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 singlevalued ==
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 termbyterm analysis of Section 7.1 above,
[157], of the
and
we
shall
see
loopexpansion
but it also character
izes the nonexactness of the DuistermaatHeckman 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 DuistermaatHeckman 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 DuistermaatHeckman 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 globallydefined 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 nontrivial odddegree can
dinate transformation in

254
Beyond the SemiClassical Approximation
7.
homology
explicitly
and
what type of flow the Hamiltonian vector field gen
erates.
Recall that the coordinate functions x" map the constant coordinate lines 2n1 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
nonintegrable 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
2ndimensional 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
equivariantlyclosed differential form (3.57). patches P, in their interior there is a welldefined (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
globallydefined realvalued C'independent of the coordinate
that it is
Killing equation on P. Thus on each coordinates, we can solve the Lie deriva
satisfies the
choice of
the
is
(7.67)
given though this cannot be extended to the whole of M. Then each integral over P in (7.66) can be written using the formula (7.46), restricted to the patch P, to get
tive
constraint,
Here
even
we assume
that M is compact, but
also be extended to the
phase
space R
2n
we
shall
see
that this formalism
can
7.3 Corrections to the DuistermaatHeckman Formula
f
(P)
T
f
here,
e'iTHsKv
ds
(n
1)!

g(VI.)
SOV)n1
(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
(766), represents
the lowestorder term in the semiclassical expansion of the partition function over M, i.e. the DuistermaatHeckman term Zo(T) in (3.63), while the
boundary terms give the general corrections to this formula and represent nontriviality that occurs rendering inexact the stationaryphase 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)n1
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
loopexpansion 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,THsKv
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
metricdependent quantities appearing
in
(7.70)
can
be written
as
91:' V.
ewP(x)_ aA Xl(x) dX
V'(X)aAX1(X)
Kv(x)lp
=
gp(V, V)
=
ewP(
(7.73)
7.
256
Beyond the SemiClassical Approximation
QVIP
ewP(II)
=
al\Xl (OAV' 2(V,\aAT_2 f
a X1

a V
+VAa,xxl (91, ppaxl ,O, op,01,xl) I dxl'
A
A0140 dx'
(7.74)
When these expressions are substituted back into the correction term (7.70), find that the integrands of JZ(T) depend only on the coordinate function
we
X1 (x). This is not surprising, since the only effect of the other coordinate functions, which define local action variables of the dynamical system, is to make the effect of the partitioning of M into patches above nontrivial, 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 DuistermaatHeckman 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 nonzero,
as
the
zeroes
of the vector field V have been
gp(V, ) thereby making it singular. We can therefore now carry out the explicit sintegral 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 stationaryphase 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 2dimensions things can be simplified rather nicely and the analysis reveals some very interesting properties of this formalism which could be generalized to higherdimensional symplectic manifolds. To start, we notice that in 2dimensions, 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 1cycles 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 sintegration we find that the MV
are
make
=
2dimensional correction terms
JZ(T)
=
1
E iT e
can
1
be written e
as
iTH(x)
V1(49,\X1(X)
49jX'(x)dxt'
(7.75)
7.3 Corrections to the DuistermaatHeckman Formula
XI,
As for the function
need to choose
we
one
which is
independent
257
of the
other coordinate transformation function X 2 to ensure that these 2 functions truly do define a (local) diffeomorphism of M. The simplest choice, as far as the evaluation of
(7.75)
is to choose
concerned,
is
x1
as
the solution of the
firstorder 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
(775)
p,
v
=
become
6Z(T) where
we
Fl,,,
2iT
(7.77)
have introduced the 1form 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 DuistermaatHeckman formula. To each of the
cycles
at E
called their Poincar6 dual
grals
of 1forms
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 SemiClassical
(iTdH +,q)
=
47r
1: PEMv
M
(_,),\(P)Fdeet (p) at 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 DuistermaatHeckman formula generate the Poincar6 duals to
homology cycles which signify that the Hamiltonian vector field does not generate a globally welldefined group action on M. When the correction the
1form
qliT
is added to the 1form 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 DuistermaatHeckman formula exact for the function. This
which defines the flow
1form is new
required for the DuistermaatHeckman 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 lowestorder term Zo(T) of its semiclassical 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
stationaryphase
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 nontrivial representative in H1 (M; R) for w (V, ) (recallq C H1 (M; R)). Thus in our approach here, the corrections to the DuistermaatHeckman formula compute (possibly) nontrivial cohomology classes of the manifold M and express geometrically what symmetry is missing from the original dynamical system that prevents its saddlepoint approximation from being exact. The explicit constructions of the Poincar6 duals above are wellknown [32] M of S1 in M which corresponds to the one takes the embedding at : S1 current which is the Dirac deltafunction DeRham. its and constructs loop at, 1form 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 1form 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 1form 0 while
Flat
1 =

V2(X)
(dxl
ing
can
thought higherloop
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 stationaryphase 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 wellacquainted. In the case of the Riemann sphere we have little to add at this point since the height function (2.1) localizes. The only point we wish
classes of
to make here is that the covariant Hessian in this
standard Kdhler geometry of S2 metric gS2 by
VVhzo
=
2
1
(I
(see
Z,
+
Z )3
Section
dz o d,
=
5.5)
2(1
case
with respect to the
is related to the Khhler

hzo)gS2
(7.86)
260
Beyond the SemiClassical Approximation
7.
which is in agreement with the analysis of Section 7.1 above. This shows the precise mechanism (i.e. the Hessian of hzo generates covariantly the Khhler
S2)
structure of
that makes the
loop
corrections vanish.
An interesting check of the above formalisms is provided by a modified version of the height function h_ro which is the quadratic functional (2)
hZo
2
hzo
=

hZ
which has the
for H
COS
2z.
0)2
I + Zf
+ Zf
hEo.
as
2zf
Now
gz (z,, )
,
Z
=
)
2
(7.87)
find that the metric
we
the isothermal solution
(2) (2) a2 h ZO /o9zh Zo
9Z9Z2
[157]
H'(zf)
(7.88)
(2)
2zf
(2) VVh Zo
(7.89)
As
_
by taking
solved
are
ZZ
(I
_
hZo ,which follows from (7.24) written in local isothermal coordinates implicitly defined metric. Thus the solution to (7.21) is
=
for the
9
0)
critical behaviour
same
equations (7.21)
F;ZZ
COS
0
(z,

+
Z )3

1) dz (9 df
=
gz2dz
does not coincide with the standard Khhler
(7.89)
0 d,
geometry of
S2,
the
partition function in this case is not exact, as Iloop 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(HH 2) e
do ei02 J ,/27ri f dlIL /2
=
M
e
i(T2iv 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,
=
(378) 
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 loopexpansion 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 DuistermaatHeckman example. This is because of the saddlepoints 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 globallydefined 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 FubiniStudy geometry defined by (7.34). In this case ^/ = 0 and
associated
(7.34)
the curvature
is trivial. The 2form Q does not determine the
same
cohomology class as the Kiihler 2form of Z' does, so that there is not enough "mixing" of the Hessian and Liouville terms in the loop expansion to cancel out higherorder corrections. For the sphere, the FubiniStudy 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
nontrivial Kiffiler structure
=
S2).
is
It is the lack
the torus here that makes
on
dynamical systems on it nonintegrable. Although the failure of the DuistermaatHeckman theorem in this case 2 via Kircan be understood in terms of the nontrivial 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 Clfunction 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 SemiClassical
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 (37r/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 equivariantlylocalizable 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, globallydefined 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 Rlaction. Thus the above formalism implies that the corrections to the DuistermaatHeckman 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 1form 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(r2rl)
iT Im
r
f
eiT
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
eiT
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
eiT
iT(r2+rl)
2
(7.98)
find
we
eiT(r2ri)
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 saddlepoint series for this Hamiltonian diverges. The exponential integral function can be expanded as the series [60] the DuistermaatHeckman formula.
and the overall
264
7.
Beyond
the SemiClassical
Approximation 00
Ei(x)
7 +
=
log x
+
n
E nn!
(7.100)
n=1
for
x
y is the EulerMascheroni constant. Thus the
small, where
divergent
1, giving a much T simpler way to read off the coefficients of the loopexpansion (note the enormous complexity of the series coefficients in (7.3) for this Hamiltonian a direct signal of the messiness of its stationaryphase 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 DuistermaatHeckman formula gave Zo Thus Zo + 6ZT2 1972.41, which is a better approximation to the partition function than the DuistermaatHeckman 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
nondegenerate
case

i9q
proceeding
as
(7101)
=
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 DuistermaatHeckman 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 DuistermaatHeckman 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 loopexpansion 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 saddlepoint expansion. Indeed, we see that it reproduces the precise analytic features of the loopexpansion 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 conformallyinvariant geometries for this dynamical system when the potential U(q) > 0 is bounded from below. In the harmonicpolar coordinates (5.274), the conformal Killing equations (7.35) can be determined by setting the righthand 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
singlevaluedness
is true, i.e. when
U(q)
=
266
with
Beyond
7.
V
=
=
Approximation
oscillator, the equations (7.106) and radiallysymmetric 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 SemiClassical
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
2loop,
(7.19) in harmonic coordinates. (7.8) has components
correction
the connection 1form
^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 (7110). 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 COOfunction on R 2. Thus the only potential which is bounded from below that leads to a localizable partition function is that of the simpleharmonic 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
semiclassical 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 finitedimensional 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 higherloop terms in the path integral loop expansion is a result of the usual nonrenormalizations of Iloop 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 nonlinear sigmamodel (see Chapter 8). Another place where the metric could enter into a breakdown of the localization formulas is when the localization 1form V) iwg does not lead to a homotopicallytrivial element under the (infinitesimal) supersymmetry transformation described by QS. Then additional input into the localization formalism should be required on a topologically nontrivial phase space to ensure that QSO indeed does reside in the trivial homotopy class. These inputs could follow from an appropriate loop space extension of the correction in Section 4.4 that the
,\ E
>


268
terms
7.
Beyond the SemiClassical Approximation
JZ(T)
discussed
above,
which will then
always reflect global properties
of the quantum theory. Other directions could also entail examining the connections between equivariant localization and other ideas we have discussed in this Book. One is the ParisiSourlas 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 NiemiTirkkonen 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 quantumintegrable bosonic theories and supersymmetric quantum field theories. This is particularly interesting from the point of view of examining corrections to the localization formulas, which in the last Chapter we have seen reflect global properties of the theory. This would be of particular interest to analyse more closely, as it could then lead to a unified description of localization in the symplectic loop space, the supersymmetric loop space and in topological quantum field theory. In this final Chapter we shall discuss some of the true field theoretical models to which the equivariant localization formalism can be applied. We
We have
that the quantum field theories which fall into this framework al
shall
see
ways
have,
gauge
as
anticipated,
symmetry
cohomological
or a
some
large symmetry
supersymmetry)
structure
on
that
group
serves
to
the space of fields that
(such
as a
provide
can
an
topological equivariant
be understood
as a
"hidden" supersymmetry of the theory. Furthermore, the configuration spaces of these models must always admit some sort of (pre)symplectic structure
properties of phase space path integrals can applied. Because of space considerations we have not attempted to give a detailed presentation here and simply present an overview of the various constructions and applications, mainly just presenting results that have been
in order that the localization
be
obtained. The interested reader is refered to the extensive list of references
Chapter for m ore details. We shall emphasize throughout this Book between the localizaformalism for dynamical systems and genuine topological quantum field tion theories. Exploring the connections between the topological field theories and more conventional physical quantum field theories will then demonstrate how the equivariant localization formalism for phase space path integrals serves as the correct arena for studying the (path integral) quantization of real physical that
are
cited
throughout
this
here the connections eluded to
systems.
R. J. Szabo: LNPm 63, pp. 269  294, 2000 © SpringerVerlag Berlin Heidelberg 2000
270
8.
Equivaxiant Localization
8.1 TwoDimensional
Between
Physical
In Section 3.8
we
first
and
Cohomological
in
Field
Theory
YangMills Theory: Equivalences Topological Gauge Theories
pointed
that, instead
out
consider the Poisson action of
of circular
actions,
one can
nonabelian Lie group acting on the phase space. Then the nonabelian 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 2dimensional QCD (equivalently the weakcoupling limit of 2dimensional pure YangMills 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 2dimensional QCD. The action for pure YangMills theory on a 2dimensional surface Zh of genus h is
Sym[A]
=
e2


f
tr
FA
(8.1)
FA
*
Eh a gauge connection of a trivial prinicipal Gbundle over Zh, FA is its curvature 2form (c.f. Section 2.4), and e2 is the coupling constant
where A is
theory. Since 0 = *FA is a scalar field in 2dimensions, (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 gvalued multiplier to write ZVh (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_'fhtr( 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
Schwarztype 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 TwoDimensional
YangMills Theory
271
The gauge invariance of the action S[o, A] appearing in (8.3) is expressed as S[glog, 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, gvalued fermion field 0", which is anticommuting 1form 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*0if *IQ, Tl} I tr
Zh
Zh
Zh
(8.4)
FaddeevPopov 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. gaugeinvariant) states of the quantum field theory. The system of fields (A,,O, 0) is the basic multiplet of cohomological YangMills 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
JAIt
=
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 (Zgradings) of the fields (A, 0, 0) are (0, 1, 2). We shall not enter here into a discussion of the physical characteristics of 2dimensional YangMills theory. It is a superrenormalizable quantum field theory which is exactly solvable and whose simplicity therefore allows one to explore the possible structures of more complicated nonabelian gauge theories such as higherdimensional cohomological field theories and other physical models such as 4dimensional 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 gaugefixed path integral (8.4) is an infinitedimensional version of the partition function in the last line of (3.125) used for nonabelian localization. Indeed, the integration over the auxilliary fermion fields 0 acts to
where
e
is
an
=
272
8.
produce
Equivariant Localization
in
Cohomological
Field
Theory
field theoretical
analog of the superI 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 Gequivariant 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 6AIi *
=

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 1form 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" YangMills theory with action (8.1) and the cohomological YangMills theory with actionf5htr iOFA which is defined essentially by the steps which lead to the nonabelian 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 2dimensional YangMills 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 2dimensional YangMills theory, such as CalegoroMoser integrable models
[56]. To carry out the localization onto MO explicitly, we choose a Ginvariant metric g on Zh and take the localization 1form 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
YangMills 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 TwoDimensional
ZZh (e 2)
2 
=
P?

e
11 a(P)22h a>O
(dim
YangMills Theory
RX)22h
e2 e
_T
273
(8.8)
AEZ'
expanding the various (Ginvariant) 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 nonpolynomial function of the coupling constant e2. This nonpolynomial 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 DuistermaatHeckman integration formula to problems with nonabelian symmetries) provides an unexpected and new insight into the structure of the partition function of 2dimensional YangMills 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 Gconnections on Zh [78, 82, 171]. This approach to 2dimensional YangMills theory has also been studied for genus h
=
0 in
[105].
The
intriguing mapping between a physical gauge theory, with propagating particlelike 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 nonlinear cousin of 2dimensional topological YangMills theory, the gauged GIG WessZuminoWitten 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(gldg)A3 M
I
tr
(A A dgg1
+ A A
(8.9)
Ag)
Zh
where g (= C, (Zh' g) is a smooth groupvalued field, M is a 3manifold with boundary the surface Eh , and A is a gauge field for the diagonal G subgroup
GR symmetry group of the ordinary (ungauged) WessZuminoby the action SG [g] SGIG [g, A 0] [54]. Since the is * when acting on 1forms, invariant Hodge duality operator conformally the action (8.9) depends only on the chosen complex structure of Z h. As for the YangMills 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 Grassmannodd variables 0 which represents the symplectic form of the
GL
x
Witten model defined
=
=
274
Equivariant Localization
8.
in
Cohomological
Field
Theory
JAJA on the space A of gauge fields on _Vh. Again the resulting theory supersymmetric and the infinitesimal supersymmetry transformations are
f,, is
JA,
=
0,
with the
6V) ,
,
=
Ag

Z
A,
;
supplemental condition Jg
JA2 =
02
=
J,02
,
0. Unlike its
=
A

(Aq)
'
Z
(8. 10)
YangMills 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 fixedpoint
localization formula is the
algebraic
Verlinde formula for the dimension of
the space of conformal blocks of the ordinary WessZuminoWitten 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
22h
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
22h
also be used to shed
(8.11)
light
on
formal structures of 2dimensional conformal field theories.
path integral for a version of the ordinary Wessgive a field theoretical generalization of Stone's derivation for the WeylKac character formula for KacMoody algebras (i.e. loop groups). This is done by exploiting the GL x GR KacMoody 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 KacMoody group d is a central extension of the loop group S' + d 4 LG of a compact semisimple 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
ZuminoWitten 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 TwoDimensional
x
along
central
the
loop
SIpart
in
Lg and the
YangMills 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
2dimensional 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 +
g1 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 WessZuminoWitten model with the single KacMoody 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 infinitedimensional
DuistermaatHeckman integration formula for this path integral. The critical points of the action (8.13) are in onetoone correspondence with the affine Weyl group Waff (Hc) W(Hc) z) kc, where Ac is the set of coroots. 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 zeropoint 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
WeylKac
Lo tr,\ eW q
character formula
1:
[138] (for
see
coincides
Section
5.1)
( 1)'(') q E,((A+p)i+(k+h)77i2pi2)/2(k+h)
wEW(Hc),??Ef1c x
e
i(A+p+(k+h)77)(H(') )
eip(H())
qn)r n>O
X
e
C >o
ia(H())
qn)
eia(H(')) qn1 (8.14)
which arises from the expansions of various quantities defined on the torus in terms of Jacobi thetafunctions. Thus infinite dimensional analogs of the
localization formulas in quantum field theory can also lead to interesting generalizations of the character formulas that the topological field theories of earlier Chapters represented.
276
8.
Equivariant Localization
Finally, localization
it is as
possible
well. For
to
Cohomological Field Theory
use some
instance,
Scsp [A]
in
k =
87r
in
of these ideas in the context of abelian
[1261
the abelian gauge
j AAdAm j 2
M
defined
on
a
3manifold 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 ChernSimons 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 87rm/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 3dimensional quantum field theory defined by (8.15). m
4
models associated with ChernSimons
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 wellknown 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 ChernsSimons action is in fact another prototype of
logical
field
theory [221.
a
Schwarztype 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 spacetime (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 spacetime 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 anticommutation 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
anticommutation 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 spacetime 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 superPoincar6 group, see [155]. We assume here that the spacetime has Minkowski signature. We shall be interested in using the relations of the superPoincar6 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 superPoincar6 group) is to use a superspace formulation. We introduce 2 Weyl spinors 0' and #& which =
=
=
==
parametrize the infinitesimal supersymmetry transformations. Then the N I supersymmetry generators in 4 dimensions can be written as a
a

Q&
.
=
Kinetic terms in the supersymmetric action ant
_
aO& are
+
'A 
0111 a
(8.17)
constructed from the covari
superderivatives a a0a
+
iZa6,PaA
,
D&
aO&
iZ.1'6,0"a4
(8.18)
278
8.
Equivariant Localization
in
Cohomological Field Theory
readily verified that with the representation (8.17) the relations of superPoincar6 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 BakerCampbellHausdorff 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 spacetime. The lengthy algorithm to construct supermultiplets associated with a given irreducible or reducible spin representation of the superPoincar6 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 spacetime 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 Ematrices
T,
T3
=
on
the
righthand
(0 1) 1
0
0
1
(8.20)
Majorana representation selects the preferred lightcone coordinates X2
XO for the translation generators P,, in (8.16) above. Different representations of the Dirac gammamatrices would then define different pre=
lightcone 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
49Xl
(8.21) precisely the (nonstandard) 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
3A
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, IXI00 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 1forms
incides with the
pertinent N provides
construction therefore
=
I
a
280
in
Equivaxiant Localization
8.
(8.22)
and
(8.23).
It is
in
Cohomological
possible decompose
now
ric quantum field theories
Field
Theory
to examine how various
supersymmet
with respect to the above equivariant cohomological structure on the space of fields. The canonical choice is the WessZumino model which is defined by the sigmamodel action T
SWZ
d3X
1 f dt
dO d
D P + W[
2
P]
(8.27)
0
where
W[4i]
is some superpotential. 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 2form 0, S Because of the boundary conditions on the fields of the theory, the supersymmetry charges, which geometrically generate translations in the chosen lightcone 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 DS2 .TV+ Q. Thus the supersymmetric model admits a loop space symplectic structure and the corresponding path integral can be written as a superloop 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
1forms, the coordinates and fields. It is only in the simplest
mechanics)
up
1forms 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 1forms. 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
infinitedimensional
etry generator
(c.f.
Section
5.2).
8.3
Supergeometry
and the
BatalinaadkinVilkovisky
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 WessZumino model (8.27)), the WessZumino model (8.27) in both 2 and 4 dimensions, and 4dimensional N 1 supersymmetric SU(N) YangMills 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
WessZumino 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 2dimensional supersymmetric nonlinear sigmaof ParisiSourlas stochastic
(i.e. the WessZumino Section).
model
model
(827)
with
a
curved target space

see
the next
8.3
Supergeometry BatalinFradkinVilkovisky
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
gaugefixing 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 antifield formalism. In this formalism, a theory with firststage 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 Schwarztype) [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 supermanifold (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 supermanifold In the
case
of the
282
8.
Equivariant Localization
in
Cohomological
Theory
Field
and reformulate the localization concepts in the more rigorous framework of supergeometry. This is particularly important for some of the other localization features of
field theories that
topological
shall discuss in the next 2
we
Sections.
First, we shall incorporate the natural geometrical objects of the BatalinVilkovisky formalism into the equivariant localization framework. The local coordinates on the supermanifold MS are denoted as ZA (xA"qA). We define a Grassmannodd degree symplectic structure on Ms by the nondegenerate odd symplectic 2form =
S21
=
The 2form
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
superfunctions
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) superfunction A(x, 77) with the
=

the
supercoordinate
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
superfunction
f A, f },,.
In
=
2f77 ,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 jiA 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
5ZA
an
A
Supergeometry
8.3
and
(MS,fll, QH)
tions of motion
:VZ
=
determine
a
biHamiltonian pair. The
Formalism
283
corresponding
equa
are
JXO) QHJ1
Generally,
BatalinFradkinVilkovisky
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 superfunction it is
ferentiation

7
1
=
TH is
an
integral
JQHJH11 structure
=
g"VVIJI 77V
(835)
of motion for the Hamiltonian system (MS, S?1 7 QH) , i.e. 0. Furthermore, TH determines the usual biHamiltonian
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 supermanifold 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)sJHw,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 antibrackets. Furthermore,
JH

w,
e
iT(H+w)sJHw,.THJ1
9QH,IH
e
11
=
JQH,
e
we
have
iT(H+w)sJHw,TffJ1
iT(H+w)sJHw,_THR11
I,
=
0
0
(837) 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 DuistermaatHeckman 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 BatalinVilkovisky procedure for the evaluation of BRST gaugefixed
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 finitedimensional 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 cnumber 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 BatalinVilkovisky formalism is applicable [149] or in the dimensional reduction mechanism of the ParisiSourlas model [136]. A generalization to more general (nonlinear) supermanifolds can be found in [175]. Nersessian has also demonstrated how to incorporate the antibracket 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 BatalinVilkovisky 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 nonabelian 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
BatalinRadkinVilkovisky
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 4dimensional topolog ical YangMills 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 qdirection in superspace and then the connection between equivariant cohomology and BatalinFradkinVilkovisky quantization of 4dimensional topological YangMills 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 4dimensional selfdual YangMills theory (i.e. FA *FA) which is intimately connected to topological YangMills theory. Finally, the incorporation of the BatalinVilkovisky 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
BatalinRadkinVilkovisky
and the
Formalism
285
provided that M has on it a Riemannian structure. on MS is given by the supersymplectic 2form
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 Iforms, with S?' &9 , are then tonian
systems
(M, wo
on
=
=
=
=
=

=
eo
0A dx,4
=
+
gI'ZVq1iDg,v
The 2forms 0' determine the
A,Bjj
=
192
,
=
g,,,Vvdxm
+
Poisson brackets
even
(VjA)[w'tv
gj,,,qmDgij'
on
+gm'
(8.40)
MS
aA
a'q1I
B
15Oqv
(8.41)
where
v,
=
a,
rN 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 biHamiltonian 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 superloop 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, S2',Hi)
have the quantum actions
[103]
T
So
dt
(01,bm

H +
177"g,,, 2
1
dt
+
2
(.f2V)1,vqI'?7'
+
Iw,,,,AI'Av
2
0
+
1Rmv,\pA"Av?7A 77
P
2
+
1g,,vFMF'
2
T
S2
1
dt
g,,,Vv&"

KV +
1 2
77A9pV
dt
+
2
(Qv)mvqlqv
0
+
I 1(S?v)tvA"Av + R,,,,\pAI'Avq'\?7 2 2
P
1 +
2
gtzvFmF' (8.43)
286
Equivariant Localization
8.
Cohomological
in
where the quantum partition functions
Zi(T)
I
=
Field
Theory
are
2n 2n FA [d 2nX1 [d 2n,7] [d F] [d A] e'si [ x,,q; 1
(8.44)
LMsOLA'Ms
and
we
have introduced
bosonic variables F11 the usual way. If H
LTdt 2'g Av

=
dx" and auxilliary anticommuting variables A" dqll to exponentiate the determinant factors in 0 (the topological limit), then the action So + 
(0
is that of the
+
I)dimensional
N
=
supersymmetric
I
1 DeRham supersymmetric sigmamodel, 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 superloop space equivariant cohomology. This has been done explicitly in [122]. The superloop space equivariant exterior derivative is
nonlinear
=
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 eQ 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 2dimensional N 1 supersymmetric sigma
the
=
by partial localization of it to a Idimensional model. This is done by breaking its (leftright) (1, 1) supersymmetry explicitly by the Hamiltonian flow. In this procedure the usual boson kinetic term (in lightcone coordinates) g,,,,a+0A,9_0v drops out. The path integral Zo(T) can be evaluated by adding an explicit gaugefixing 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 WessZumino 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
MathaiQuillen Formalism
onto the
287
Tperiodic 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 GaussBonnetChern 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 Tperiodic classical trajectories for (timedependent) classical Hamiltonians [119, 122]. Thus, the path integrals associated with the supermanifolds defined by the BatalinVilkovisky formalism for dynamical systems lead to loop space and equivariant generalizations of other familiar topological invariants. Furthermore, these models are closely related to supersymmetric nonlinear sigmamode 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 MathaiQuillen 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 sigmamodels). Conversely, in Section 8.2 we demonstrated that arbitrary field theoretical models of these types could be placed into the loop space equivariant localization framework so that there now
calization
288
is
Equivariant Localization
8.
in
Cohomological
Field
Theory
equivalence between dynamical systems and field theory modgeometric context. The discussion of Section 8.1 then illustrated genuine, geometrical equivalences between physical and topological
sort of
a
els in this certain
gauge theories which demonstrates the power of the formalisms of both topological field theory and equivariant localization of path integrals in describing
the quantum characteristics of physical systems There is one final step for this connection which is the .
AtiyahJeffrey
geometric interpretation of generic cohomological field theories [10] which is based on the MathaiQuillen construction of Gaussianshaped 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 MathaiQuillen formalism in topological field theory can be found in [27, 29, 34]. The basic idea behind the MathaiQuillen formalism is the relation (3.71) between the Poincar6Hopf and GaussBonnetChern 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,
=
MathaiQuillen Formalism
r,\ /J,VP,\77V P, +
1
IJA

2
tZPA
V
P
289

77 77A
(8.51)
r,\ ?7vq,\ /'tV
coincides with the standard infinitesimal transformation laws of N
=
1 DeR
ham supersymmetric quantum mechanics. We now consider the following integral Z*
f
=
d
2n
d2n p d2n 77
x
d2nq
e
dO[x,p;?j,fj]
(8.52)
S*MOT*(S*M)
Since d 2= 0 the
integral (8.52) is formally independent of the function '0 on 0. We can therefore evaluate supermanifold 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 deltafunction 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
MathaiQuillen
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
socalled
Gaussianshaped
+
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 Gbundle
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 Gequivariant 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
MathaiQuillen representative general V, integrating out the Grassmarm variables 2mform, 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
Poincar6Hopf
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 GaussBonnetChern 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 MathaiQuillen formalism. For this one considers the usual Cartan equivariant exterior derivative DV and the associated Lie derivative D'V on the supermanifold 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
Poincar6dual 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
MathaiQuillen Formalism for InfiniteDimensional 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 MathaiQuillen 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 nondegenerate cases are also possible to treat in this way [121].
which
MathaiQuillen Formalism
8.5 The
for InfiniteDimensional 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
MathaiQuillen
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 finitedimensional
so
that the
localization makes these quantities welldefined. Although these numbers are not independent of V as in the finitedimensional cases, they are naturally associated with M for certain choices. The functional
integrals
which arise in
this way are equivalent to ordinary finitedimensional 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 infinitedimensional MathaiQuillen formalism The
LM
+
[34].
simplest example M
over a
this bundle is
The
regularized Euler
number of the
loop
space
V'(t).
loop space version of the integral (8.52) is a superloop space, and we replace the extederivative d there by the equivariant, loop space,one Q& on L(S*M). Lie derivative L& Q? as before is the generator of time translations,
path integral rior
is the
manifold M. The canonical vector field associated with
over
Now the
an
extended
=
X
292
8.
Equivariant Localization
in
Cohomological
Field
Theory
and
employing the standard conjugation above the path integral can be localized using any singlevalued 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
finitedimensional 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 sigmamodel actually has 2 independent (holomorphic and antiholomorphic) supersymmetries. Restricting the computation of the supersymmetric quantum mechanics partition function to the antiholomorphic 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 righthand side of (8.47) coincides with the lefthand 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 finitedimensional relation (8.60). These relations play a deeper role when the Hamiltonian depends explicitly on time t. Then the righthand 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 nonlinear sigmamodels. In [124], these relations were related to a functional Euler character in the quantum cohomology defined by the topological nonlinear sigmamodel 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 infinitedimensional space
AIG
of gauge connections modulo gauge transformations of a principal Gbundle. 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
MathaiQuillen Formalism for InfiniteDimensional Vector Bundles
293
by applying the MathaiQuillen 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 4manifolds. Topological YangMills theory in 4dimensions is another in
teresting application of this formalism. The field theoretic generalization of supersymmetric quantum mechanics, i.e. the topological sigmamodel [22], is the appropriate setting for studying the quantum symmetries of string theory and more generally superconformal field theories. The MathaiQuillen formalism applied to 2dimensional topological gravity could presumably shed light on its equivalence with physical gravity in 2 dimensions. The coupling of the topological sigmamodel to topological gravity can be interpreted as topological string theory and studied using these methods. Finally, viewing 2dimensional YangMills 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 YangMills theory [34]. Thus, the MathaiQuillen 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 stationaryphase formula) and the constructive MathaiQuillen 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 2dimensional
a
Grassmannodd 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 5action is not free and that the path integral receives contributions from some arbitrarily small Jinvariant 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 MathaiQuillen formalism which shows how such infinitedimensional integrations are a pTioTi designed to represent finitedimensional integrals. The =
294
8.
Equivariant Localization
in
Cohomological
Field
Theory
antibracket formalism
developed in Section 8.3 above is a key stepping stone MathaiQuillen 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 4dimensional topological YangMill 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 Rmatrix formulations and the YangBaxter 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 YangMills theory as a useful device for proving the renormalizability of nonabelian gauge theories in 4 dimensions. It
was
shown that
present after YangMills 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 FaddeevPopov ghost and antighost fields 0a, #a a0a anticommuting Grassmann variables that transform in the adjoint representation of g. They have the canonical anticommutator
Introduce

[0a, Ob 1+ We define the
ghost number operator U
whose We
eigenvalues now
are
=
=
jab
(9.2)
as
Oap
integers running from
(9.3) 0 to dim g.
introduce the operator
Q
=
OaKa
1 
2
f abcoaob c
(9.4)
as the BRST charge, while algebra coboundary operator that computes the cohomology of the Lie algebra g with values in the representation defined by the operators Ka. The crucial property of Q is that it is 0, which can be seen from (9.1) and the identity nilpotent, Q2
In the
physics literature the operator Q
is known
in the mathematics literature it is the Lie
=
R. J. Szabo: LNPm 63, pp. 295  298, 2000 © SpringerVerlag Berlin Heidelberg 2000
296
9.
Appendix A:
BRST
f abef
Quantization
cde
f bdcf
+
cae
+
f dacf
cbe
(9.5)
0
=
which follows from
(9.1) via the Jacobi identity for the Lie bracket. Let Rk be the Hilbert space of states of ghost number k, i.e. UTI k T1 for T1 E I& We say that a state T1 E Hk is BRST invariant if it is annihilated by Q, =
QT1
0, where
=
number
by
general
in
Any
1.
the action of
other state TV
T1 +
=
Q QX

any state raises the ghost ghost number k is regarded X E Hk1. 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 BRSTcohomology in the physics literature. Mathematically, it forms the kth 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 BRSTinvariant
are
state TV of
antighost 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 BRSTinvariant if and only if it ginvariant, and thus the cohomology group HI(g; R) coincides with the space of ginvariant 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 gaugefixing 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 Ginvariant gauge field action and the volume factor vol(G) is infinite for a local gauge field
fXal
=
theory. Modulo this infinite factor, the righthand side of (9.8) is what is taken as the definition of the quantum gauge theory partition function. Introducing FaddeevPopov ghost fields and additional auxilliary fields 0a, we can absorb
Appendix
9.
the additional factors
on
the
righthand
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
+
gaugefixed, quantum action. BRSTsymmetry 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 scalarvalued 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
gaugefixed field theory, and the statement that the partition function (9.8) is independent of the choice of gaugefixing functions ga is equivalent to the fact that the path integral depends only on the BRSTcohomology 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. gaugeinvariant)
=
SO

(1)POS
(9.13)
ghostdegree 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, BRSTcohomology is the fundamental structure in topological field theories [22]. By definition, a topological action is a Wittentype action if the the classical action S is BRSTexact, while it is a Schwarztype action if the gaugefixed, quantum action S. is
(9.8)
and
(9.12)
demonstrate the
and BRST
BRSTexact
(but
not the classical
one).
In the
case
of the localization for
BRSToperator 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 BRSTsupersymmetry 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 firstclass 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
Appendix we shall briefly outline the Gequivariant cohomology of
In this
els for
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
by
terized
the fact that it is contractible and that G acts
it without fixed
on
is called the universal Gbundle. The
classifying
space points. The space EG BG for Gbundles 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 Gbundle E + M + BG. The over a manifold M has an isomorphic copy sitting inside EG >
isomorphism classes of principal Gbundles are therefore in onetoone 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
while for G and
Bu(j)
gauge
H*(BG)Z,
example, when G
For
=
=
U(1) CP 00
get
we
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 YangMills potentials
n= 0
so
=
R. J. Szabo: LNPm 63, pp. 299  308, 2000 © SpringerVerlag Berlin Heidelberg 2000
300
Appendix
10.
B: Other Models of
Equivariant Cohomology
while BG AIG is the space of gauge orbits. In string theory G is the semidirect product of the difleomorphism and Weyl groups of a Riemann surface _Th of genus h, EG is the space of metrics on _T h, and BG is the moduli =
space MEh of _Th From this point of view, one can define topological field theory and topological string theory as the study of H*(BG) and related cohomologies using the language of local quantum field theory. .
Given and M M
x
a
on
smooth Gaction
M, on
we
thus have 2 spaces EG
the Cartesian
product
space
Eg via the diagonal action G
x
(M
Like the Gaction
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 Gaction 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)
Gequivariant 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 nonabelian lo
topological field theories
and
integrable models
very
powerful techniques.
10.2 The Weil Model The topological definition of the Gequivariant 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 anticommuting
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 1forms Oa the i.e. generators Oa of S(g*) have degree 2 Zgrading (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) nonvanishing 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
dpvoa
7
all
are
d2W
Equivariant Cohomology
exterior derivatives
_fabcoboc
nilpotent derivations
d20
=
=
=
2
d
=
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 nontrivial 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 Gbundle
E
the curvature 2form 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 1form 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 ChernWeil 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 Gbundle EG. Like EG, it possesses universal properties, and therefore it provides a universal model of connections on Gbundles. 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 nontrivial and universal cohomology classes by considering the socalled "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 Gaction 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 Gaction on it, and the ChernWeil 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 Ginvariant (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) Gequivariant 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 ChernWeil
+
of
k