Introduction and Overview&E.Bick,F.D.Steffen 1
1 Topology and Geometry in Physics 1
2 An Outline of the Book 2
3 Complementary Literature 4
Topological Concepts in Gauge Theories&F.Lenz 7
1 Introduction 7
2 Nielsen-Olesen Vortex 9
2.1 Abelian Higgs Model 9
2.2 Topological Excitations 14
3 Homotopy 19
3.1 The Fundamental Group 19
3.2 Higher Homotopy Groups 24
3.3 Quotient Spaces 26
3.4 Degree of Maps 27
3.5 Topological Groups 29
3.6 Transformation Groups 32
3.7 Defects in Ordered Media 34
4 Yang-Mills Theory 38
5 't Hooft-Polyakov Monopole 43
5.1 Non-Abelian Higgs Model 43
5.2 The Higgs Phase 45
5.3 Topological Excitations 47
6 Quantization of Yang-Mills Theory 51
7 Instantons 55
7.1 Vacuum Degeneracy 55
7.2 Tunneling 56
7.3 Fermions in Topologically Non-trivial Gauge Fields 58
7.4 Instanton Gas 60
7.5 Topological Charge and Link Invariants 62
8 Center Symmetry and Confinement 64
8.1 Gauge Fields at Finite Temperature and Finite Extension 65
8.2 Residual Gauge Symmetries in QED 66
8.3 Center Symmetry in SU(2)Yang-Mills Theory 69
8.4 Center Vortices 71
8.5 The Spectrum of the SU(2) Yang-Mills Theory 74
9 QCD in Axial Gauge 76
9.1 Gauge Fixing 76
9.2 Perturbation Theory in the Center-Symmetric Phase 79
9.3 Polyakov Loops in the Plasma Phase 83
9.4 Monopoles 86
9.5 Monopoles and Instantons 89
9.6 Elements of Monopole Dynamics 90
9.7 Monopoles in Diagonalization Gauges 91
10 Conclusions 93
Aspects of BRST Quantization&J.W.van Holten 99
1 Symmetries and Constraints 99
1.1 Dynamical Systems with Constraints 100
1.2 Symmetries and Noether's Theorems 105
1.3 Canonical Formalism 109
1.4 Quantum Dynamics 113
1.5 The Relativistic Particle 115
1.6 The Electro-magnetic Field 119
1.7 Yang-Mills Theory 121
1.8 The Relativistic String 124
2 Canonical BRST Construction 126
2.1 Grassmann Variables 127
2.2 Classical BRST Transformations 130
2.3 Examples 133
2.4 Quantum BRST Cohomology 135
2.5 BRST-Hodge Decomposition of States 138
2.6 BRST Operator Cohomology 142
2.7 Lie-Algebra Cohomology 143
3 Action Formalism 146
3.1 BRST Invariance from Hamilton's Principle 146
3.2 Examples 147
3.3 Lagrangean BRST Formalism 148
3.4 The Master Equation 152
3.5 Path-Integral Quantization 154
4 Applications of BRST Methods 156
4.1 BRST Field Theory 156
4.2 Anomalies and BRST Cohomology 158
Appendix Conventions 165
Chiral Anomalies and Topology&J.Zinn-Justin 167
1 Symmetries,Regularization,Anomalies 167
2 Momentum Cut-Off Regularization 170
2.1 Matter Fields：Propagator Modification 170
2.2 Regulator Fields 173
2.3 Abelian Gauge Theory 174
2.4 Non-Abelian Gauge Theories 177
3 Other Regularization Schemes 178
3.1 Dimensional Regularization 179
3.2 Lattice Regularization 180
3.3 Boson Field Theories 181
3.4 Fermions and the Doubling Problem 182
4 The Abelian Anomaly 184
4.1 Abelian Axial Current and Abelian Vector Gauge Fields 184
4.2 Explicit Calculation 188
4.3 Two Dimensions 194
4.4 Non-Abelian Vector Gauge Fields and Abelian Axial Current 195
4.5 Anomaly and Eigenvalues of the Dirac Operator 196
5 Instantons,Anomalies,and 0-Vacua 198
5.1 The Periodic Cosine Potential 199
5.2 Instantons and Anomaly：CP(N-1) Models 201
5.3 Instantons and Anomaly：Non-Abelian Gauge Theories 206
5.4 Fermions in an Instanton Background 210
6 Non-Abelian Anomaly 212
6.1 General Axial Current 212
6.2 Obstruction to Gauge Invariance 214
6.3Wess-Zumino Consistency Conditions 215
7 Lattice Fermions：Ginsparg-Wilson Relation 216
7.1 Chiral Symmetry and Index 217
7.2 Explicit Construction：Overlap Fermions 221
8 Supersymmetric Quantum Mechanics and Domain Wall Fermions 222
8.1 Supersymmetric Quantum Mechanics 222
8.2 Field Theory in Two Dimensions 226
8.3 Domain Wall Fermions 227
Appendix A．Trace Formula for Periodic Potentials 229
Appendix B．Resolvent of the Hamiltonian in Supersymmetric QM 231
Supersymmetric Solitons and Topology&M.Shifman 237
1 Introduction 237
2 D=1+1;N=1 238
2.1 Critical(BPS)Kinks 242
2.2 The Kink Mass(Classical) 243
2.3 Interpretation of the BPS Equations Morse Theory 244
2.4 Quantization.Zero Modes：Bosonic and Fermionic 245
2.5 Cancelation of Nonzero Modes 248
2.6 Anomaly Ⅰ 250
2.7 Anomaly Ⅱ(Shortening Supermultiplet Down to One State) 252
3 Domain Walls in(3+1)-Dimensional Theories 254
3.1 Superspace and Superfields 254
3.2 Wess-Zumino Models 256
3.3 Critical Domain Walls 258
3.4 Finding the Solution to the BPS Equation 261
3.5 Does the BPS Equation Follow from the Second Order Equation of Motion? 261
3.6 Living on a Wall 262
4 Extended Supersymmetry in Two Dimensions：The Supersymmetric CP(1) Model 263
4.1 Twisted Mass 266
4.2 BPS Solitons at the Classical Level 267
4.3 Quantization of the Bosonic Moduli 269
4.4 The Soliton Mass and Holomorphy 271
4.5 Switching On Fermions 273
4.6 Combining Bosonic and Fermionic Moduli 274
5 Conclusions 275
Appendix A．CP(1)Model=O(3)Model(N=1 Superfields N) 275
Appendix B．Getting Started(Supersymmetry for Beginners) 277
B.1 Promises of Supersymmetry 280
B.2 Cosmological Term 281
B.3 Hierarchy Problem 281
Forces from Connes'Geometry&T.Schücker 285
1 Introduction 285
2 Gravity from Riemannian Geometry 287
2.1 First Stroke：Kinematics 287
2.2 Second Stroke：Dynamics 287
3 Slot Machines and the Standard Model 289
3.1 Input 290
3.2 Rules 292
3.3 The Winner 296
3.4 Wick Rotation 300
4 Connes'Noncommutative Geometry 303
4.1 Motivation：Quantum Mechanics 303
4.2 The Calibrating Example：Riemannian Spin Geometry 305
4.3 Spin Groups 308
5 The Spectral Action 311
5.1 Repeating Einstein's Derivation in the Commutative Case 311
5.2 Almost Commutative Geometry 314
5.3 The Minimax Example 317
5.4 A Central Extension 322
6 Connes'Do-It-Yourself Kit 323
6.1 Input 323
6.2 Output 327
6.3 The Standard Model 329
6.4 Beyond the Standard Model 337
7 Outlook and Conclusion 338
Appendix 340
A.1 Groups 340
A.2 Group Representations 342
A.3 Semi-Direct Product and Poincaré Group 344
A.4 Algebras 344
Index 351