Equivariant cohomology and localization of path integrals

Cover ......Page 1
Lecture Notes in Physics: Monographs 63 ......Page 2
Equivariant Cohomology and Localization of Path Integrals ......Page 4
Preface ......Page 6
Contents ......Page 8
1.1 Path Integrals in Quantum Mechanics, Integrable Models and Topological Field Theory ......Page 12
1.2 Equivariant Localization Theory ......Page 16
1.3 Outline ......Page 18
2.1 Example: The Height Function of the Sphere ......Page 22
2.2 A Brief Review of DeRham Cohomology ......Page 24
2.3 The Cartan Model of Equivariant Cohomology ......Page 30
2.4 Fiber Bundles and Equivariant Characteristic Classes ......Page 35
2.5 The Equivariant Localization Principle ......Page 44
2.6 The Berline-Vergne Theorem ......Page 49
3. Finite-Dimensional Localization Theory for Dynamical Systems ......Page 54
3.1 Symplectic Geometry ......Page 55
3.2 Equivariant Cohornology on Symplectic Manifolds ......Page 58
3.3 Stationary-Phase Approximation and the Duistermaat-Heckman Theorem ......Page 62
3.4 Morse Theory and Kirwan\'s Theorem ......Page 67
3.5 Examples: The Height Rinction of a Riemann Surface ......Page 69
3.6 Equivariant Localization and Classical Integrability ......Page 73
3.7 Degenerate Version of the Duistermaat-Heckman Theorem ......Page 78
3.8 The Witten Localization Formula ......Page 81
3.9 The Wu Localization Formula ......Page 85
4. Quantum Localization Theory for Phase Space Path Integrals ......Page 88
4.1 Phase Space Path Integrals ......Page 89
4.2 Example: Path Integral Derivation of the Atiyah-Singer Index Theorem \' ......Page 95
4.3 Loop Space Symplectic Geometry and Equivariant Cohomology ......Page 106
4.4 Hidden Supersymmetry and the Loop Space Localization Principle ......Page 110
4.5 The WKB Localization Formula ......Page 116
4.6 Degenerate Path Integrals and the Niemi-Tirkkonen Localization Formula ......Page 118
4.7 Connections with the Duistermaat-Heckman Integration Formula ......Page 123
4.8 Equivariant Localization and Quantum Integrability ......Page 125
4.9 Localization for Functionals of Isometry Generators ......Page 128
4.10 Topological Quantum Field Theories ......Page 132
5. Equivariant Localization on Simply Connected Phase Spaces: Applications to Quantum Mechanics, Group Theory and Spin Systems ......Page 138
5.1 Coadjoint Orbit Quantization and Character Formulas ......Page 141
5.2 Isometry Groups of Simply Connected Riemannian Spaces ......Page 148
5.3 Euclidean Phase Spaces and Holornorphic Quantization ......Page 157
5.4 Coherent States on Homogeneous Kdhler Manifolds and Holomorphic Localization Formulas ......Page 164
5.5 Spherical Phase Spaces and Quantization of Spin Systems ......Page 169
5.6 Hyperbolic Phase Spaces ......Page 180
5.7 Localization of Generalized Spin Models and Hamiltonian Reduction ......Page 182
5.8 Quantization of Isospin Systems ......Page 191
5.9 Quantization on Non-Homogeneous Phase Spaces ......Page 202
6. Equivariant Localization on Multiply Connected Phase Spaces: Applications to Homology and Modular Representations ......Page 214
6.1 Isometry Groups of Multiply Connected Spaces ......Page 216
6.2 Equivariant Hamiltonian Systems in Genus One ......Page 218
6.3 Homology Representations and Topological Quantum Field Theory ......Page 221
6.4 Integrability Properties and Localization Formulas ......Page 224
6.5 Holomorphic Quantization and Non-Symmetric Coadjoint Orbits ......Page 228
6.6 Generalization to Hyperbolic Riemann Surfaces ......Page 237
7. Beyond the Semi-Classical Approximation ......Page 244
7.1 Geometrical Characterizations of the Loop Expansion ......Page 245
7.2 Conformal and Geodetic Localization Symmetries ......Page 255
7.3 Corrections to the Duistermaat-Heckman Formula: A Geometric Approach ......Page 264
7.4 Examples ......Page 270
7.5 Heuristic Generalizations to Path Integrals: Supersymmetry Breaking ......Page 277
8. Equivariant Localization in Cohomological Field Theory ......Page 280
8.1 Two-Dimensional Yang-Mills Theory: Equivalences Between Physical and Topological Gauge Theories ......Page 281
8.2 Symplectic Geometry of Poincare Supersymmetric Quantum Field Theories ......Page 287
8.3 Supergeometry and the Batalin-Fradkin-Vilkovisky Formalism ......Page 292
8.4 Equivariant Euler Numbers, Thom Classes and the Mathai-Quillen Formalism ......Page 298
8.5 The Mathai-Quillen Formalism for Infinite-Dimensional Vector Bundles ......Page 302
9. Appendix A: BRST Quantization ......Page 306
10.1 The Topological Definition ......Page 310
10.2 The Weil Model ......Page 311
10.3 The BRST Model ......Page 315
10.4 Loop Space Extensions ......Page 317
References ......Page 320