ISBN: 3-540-67126-9
TITLE: Equivariant Cohomology and Localization of Path Integrals
AUTHOR: Szabo, Richard J.
TOC:

1. Introduction 1
1.1 Path Integrals in Quantum Mechanics, Integrable Models and Topological Field Theory 1
1.2 Equivariant Localization Theory 5
1.3 Outline 7
2. Equivariant Cohomology and the Localization Principle 11
2.1 Example: The Height Function of the Sphere 11
2.2 A Brief Review of DeRham Cohomology 13
2.3 The Cartan Model of Equivariant Cohomology 19
2.4 Fiber Bundles and Equivariant Characteristic Classes 24
2.5 The Equivariant Localization Principle 33
2.6 The Berline-Vergne Theorem 38
3. Finite-Dimensional Localization Theory for Dynamical Systems 43
3.1 Symplectic Geometry 44
3.2 Equivariant Cohomology on Symplectic Manifolds 47
3.3 Stationary-Phase Approximation and the Duistermaat-Heckman Theorem 51
3.4 Morse Theory and Kirwan's Theorem 56
3.5 Examples: The Height Function of a Riemann Surface 58
3.6 Equivariant Localization and Classical Integrability 62
3.7 Degenerate Version of the Duistermaat-Heckman Theorem 67
3.8 The Witten Localization Formula 70
3.9 The Wu Localization Formula 74
4. Quantum Localization Theory for Phase Space Path Integrals 77
4.1 Phase Space Path Integrals 78
4.2 Example: Path Integral Derivation of the Atiyah-Singer Index Theorem 84
4.3 Loop Space Symplectic Geometry and Equivariant Cohomology 95
4.4 Hidden Supersymmetry and the Loop Space Localization Principle 99
4.5 The WKB Localization Formula 105
4.6 Degenerate Path Integrals and the Niemi-Tirkkonen Localization Formula 107
4.7 Connections with the Duistermaat-Heckman Integration Formula 112
4.8 Equivariant Localization and Quantum Integrability 114
4.9 Localization for Functionals of Isometry Generators 117
4.10 Topological Quantum Field Theories 121
5. Equivariant Localization on Simply Connected Phase Spaces: Applicationsto Quantum Mechanics, Group Theory and Spin Systems 127
5.1 Coadjoint Orbit Quantization and Character Formulas 130
5.2 Isometry Groups of Simply Connected Riemannian Spaces 137
5.3 Euclidean Phase Spaces and Holomorphic Quantization 146
5.4 Coherent States on Homogeneous Khler Manifolds and Holomorphic Localization Formulas 153
5.5 Spherical Phase Spaces and Quantization of Spin Systems 158
5.6 Hyperbolic Phase Spaces 169
5.7 Localization of Generalized Spin Models and Hamiltonian Reduction 171
5.8 Quantization of Isospin Systems 180
5.9 Quantization on Non-Homogeneous Phase Spaces 191
6. Equivariant Localization on Multiply Connected Phase Spaces: Applications to Homology and Modular Representations 203
6.1 Isometry Groups of Multiply Connected Spaces 205
6.2 Equivariant Hamiltonian Systems in Genus One 207
6.3 Homology Representations and Topological Quantum Field Theory 210
6.4 Integrability Properties and Localization Formulas 213
6.5 Holomorphic Quantization and Non-Symmetric Coadjoint Orbits 217
6.6 Generalization to Hyperbolic Riemann Surfaces 226
7. Beyond the Semi-Classical Approximation 233
7.1 Geometrical Characterizations of the Loop Expansion 234
7.2 Conformal and Geodetic Localization Symmetries 244
7.3 Corrections to the Duistermaat-Heckman Formula: A Geometric Approach 253
7.4 Examples 259
7.5 Heuristic Generalizations to Path Integrals: Supersymmetry Breaking 266
8. Equivariant Localization in Cohomological Field Theory 269
8.1 Two-Dimensional Yang-Mills Theory: Equivalences Between Physical and Topological Gauge Theories 270
8.2 Symplectic Geometry of Poincar Supersymmetric Quantum Field Theories 276
8.3 Supergeometry and the Batalin-Fradkin-Vilkovisky Formalism 281
8.4 Equivariant Euler Numbers, Thom Classes and the Mathai-Quillen Formalism 287
8.5 The Mathai-Quillen Formalism for Infinite-Dimensional Vector Bundles 291
9. Appendix A: BRST Quantization 295
10. Appendix B: Other Models of Equivariant Cohomology 299
10.1 The Topological Definition 299
10.2 The Weil Model 300
10.3 The BRST Model 304
10.4 Loop Space Extensions 306
References 309
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