Topics in Physical Mathematics (eBook)

Contents 8
Preface 12
Acknowledgements 22
Chapter 1:Algebra 24
1.1 Introduction 24
1.2 Algebras 25
1.2.1 Graded Algebras 31
1.3 Kac--Moody Algebras 33
1.4 Clifford Algebras 37
1.5 Hopf Algebras 43
1.5.1 Quantum Groups 44
1.6 Monstrous Moonshine 44
1.6.1 Finite Simple Groups 46
1.6.2 Modular Groups and Modular Functions 49
1.6.3 The monster and the Moonshine Conjectures 51
Chapter 2:Topology 56
2.1 Introduction 56
2.2 Point Set Topology 58
2.3 Homotopy Groups 61
2.3.1 Bott Periodicity 72
2.4 Singular Homology and Cohomology 73
2.5 de Rham Cohomology 81
2.5.1 The Intersection Form 83
2.6 Topological Manifolds 84
2.6.1 Topology of 2-Manifolds 84
2.6.2 Topology of 3-Manifolds 85
2.6.3 Topology of 4-manifolds 87
2.7 The Hopf Invariant 91
2.7.1 Kervaire invariant 93
Chapter 3:Manifolds 95
3.1 Introduction 95
3.2 Differential Manifolds 96
3.3 Tensors and Differential Forms 104
3.4 Pseudo-Riemannian Manifolds 110
3.5 Symplectic Manifolds 114
3.6 Lie Groups 117
Chapter 4:Bundles and Connections 128
4.1 Introduction 128
4.2 Principal Bundles 129
4.3 Associated Bundles 137
4.4 Connections and Curvature 140
4.4.1 Universal Connections 146
4.5 Covariant Derivative 148
4.6 Linear Connections 151
4.7 Generalized Connections 156
Chapter 5:Characteristic Classes 158
5.1 Introduction 158
5.2 Classifying Spaces 159
5.3 Characteristic Classes 160
5.3.1 Secondary Characteristic Classes 174
5.4 K-theory 178
5.5 Index Theorems 185
Chapter 6:Theory of Fields, I: Classical 189
6.1 Introduction 189
6.2 Physical Background 190
6.3 Gauge Fields 199
6.4 The Space of Gauge Potentials 205
6.5 Gribov Ambiguity 212
6.6 Matter Fields 216
6.7 Gravitational Field Equations 220
6.8 Geometrization Conjecture and Gravity 224
Chapter 7:Theory of Fields, II: Quantum and Topological 227
7.1 Introduction 227
7.2 Non-perturbative Methods 228
7.3 Semiclassical Approximation 236
7.3.1 Zeta Function Regularization 237
7.3.2 Heat Kernel Regularization 238
7.4 Topological Classical Field Theories (TCFTs) 240
7.4.1 Donaldson Invariants 242
7.4.2 Topological Gravity 243
7.4.3 Chern--Simons (CS) Theory 244
7.5 Topological Quantum Field Theories (TQFTs) 245
7.5.1 Atiyah--Segal Axioms for TQFT 252
Chapter 8:Yang--Mills--Higgs Fields 255
8.1 Introduction 255
8.2 Electromagnetic Fields 256
8.2.1 Motion in an Electromagnetic Field 259
8.2.2 The Bohm--Aharonov Effect 262
8.3 Yang--Mills Fields 264
8.4 Non-dual Solutions 272
8.5 Yang--Mills--Higgs Fields 275
8.5.1 Monopoles 277
8.6 Spontaneous Symmetry Breaking 279
8.7 Electroweak Theory 283
8.7.1 The Standard Model 288
8.8 Invariant Connections 291
Chapter 9:4-Manifold Invariants 295
9.1 Introduction 295
9.2 Moduli Spaces of Instantons 296
9.2.1 Atiyah--Jones Conjecture 303
9.3 Topology and Geometry of Moduli Spaces 310
9.3.1 Geometry of Moduli Spaces 314
9.4 Donaldson Polynomials 316
9.4.1 Structure of Polynomial Invariants 321
9.4.2 Relative Invariants and Gluing 323
9.5 Seiberg--Witten Theory 325
9.5.1 Spin Structures and Dirac Operators 326
9.5.2 The Seiberg--Witten (SW) Invariants 327
9.6 Relation between SW and Donaldson Invariants 330
9.6.1 Property P Conjecture 331
Chapter 10:3-Manifold Invariants 333
10.1 Introduction 333
10.2 Witten Complex and Morse Theory 334
10.3 Chern--Simons Theory 339
10.4 Casson Invariant 345
10.5 Floer Homology 346
10.6 Integer-Graded Instanton Homology 353
10.7 WRT Invariants 358
10.7.1 CFT Approach to WRT Invariants 360
10.7.2 WRT Invariants via Quantum Groups 362
10.8 Chern--Simons and String Theory 365
10.8.1 WRT Invariants and String Amplitudes 367
Chapter 11:Knot and Link Invariants 371
11.1 Introduction 371
11.2 Invariants of Knots and Links 372
11.3 TQFT Approach to Knot Invariants 382
11.4 Vassiliev Invariants of Singular Knots 387
11.5 Self-linking Invariants of Knots 388
11.6 Categorification of the Jones Polynomial 391
Epilogue 396
Appendix A:Correlation of Terminology 398
Appendix B:Background Notes 400
Appendix C:Categories and Chain Complexes 411
Appendix D:Operator Theory 420
References 435
Index 451