Geometry and quantum physics

Preface......Page 1
Contents......Page 3
1 Introduction......Page 4
2.1 Stationary Phase Method......Page 5
2.2 Equivariant Cohomology......Page 7
2.3 Proof of Localization Formula......Page 11
2.4 Duistermaat-Heckman Formula......Page 12
3.1 Weil Algebra and Weil Di erential......Page 14
3.2 Weil Model of Equivariant Cohomology......Page 16
4.1 Non-commutative Weil Algebra......Page 20
4.2 Group-Valued Moment Maps......Page 22
4.3 Group-Valued Localization......Page 23
References......Page 26
1 Introduction......Page 28
2 BF Theory: Classical Field Equations......Page 30
3 Classical Phase Space......Page 33
4 Canonical Quantization......Page 35
5 Observables......Page 43
6 Canonical Quantization via Triangulations......Page 47
7 Dynamics......Page 54
8 Spin Foams......Page 65
9 q-Deformation and the Cosmological Constant......Page 71
10 4-Dimensional Quantum Gravity......Page 76
Appendix: Piecewise Linear Cell Complexes......Page 82
2 BF Theory: Classical Field Equations......Page 83
4 Canonical Quantization......Page 84
5 Observables......Page 87
6 Canonical Quantization via Triangulations......Page 88
7 Dynamics......Page 89
8 Spin Foams......Page 90
9 q-Deformation and the Cosmological Constant......Page 91
10 4-Dimensional Quantum Gravity......Page 93
Appendix......Page 96
1 Introduction......Page 97
2 D-Branes and T-Duality......Page 101
3 R.R Charged D-Branes......Page 106
4 Microscopic Dp-Brane String Amplitudes and Metrics......Page 110
6 Near Horizon Geometry......Page 116
7 Supersymmetry......Page 119
8 T-Duality and Near Horizon Geometries......Page 123
References......Page 131
Foreword......Page 133
1.1 The Dirac Operator of a Riemannian Spin Manifold......Page 135
1.2 Spectral Triples......Page 136
1.3 Formal Forms......Page 137
1.4 Quantum DeRham Complex......Page 138
1.5 Quantum Volume Form......Page 139
1.6 Quantum Connections and Curvature......Page 140
1.7 The Quantum Yang-Mills Algorithm (for a Given Even d-Dimensional Spectral Triple (AI; H; D))......Page 142
1.8 The Electrodynamics Case......Page 143
1.9 The Two-Point Algebra CI   CI (Embryonal Higgs)......Page 144
1.10 De nition-Lemma......Page 145
2 The Electroweak Inner Spectral Triple......Page 146
2.2 The Spectral Triple (Aew; Hf ; Df )......Page 147
2.3 The Aew-Bimodule  DqA 1 of Dq-Quantum One-Forms......Page 150
2.4 The Aew-Bimodule  DqA 2 of Dq-Quantum Two-Forms......Page 153
2.5 Connections. Curvature. Yang-Mills Action......Page 159
2.7 Lemma......Page 160
2.8 Proposition......Page 161
2.9 Conclusion......Page 162
3.1 Sketch of the Connes-Lott Model......Page 163
3.2 De nition (Metric Dual Pairs)......Page 165
3.4 The Higgs Mass......Page 166
4.1 Defnition......Page 167
4.3 Proposition (The Classical Case)......Page 168
4.5 Lemma......Page 169
4.8 Lemma......Page 170
4.9 Proposition (Tensor Product of Real Spectral Triple......Page 171
4.10 De nitions......Page 172
4.11 Proposition......Page 173
4.13 Remark......Page 175
5.1 Reminder. The Inner Real Metric Dual Pair......Page 176
5.2 Definition (The Inner-Space S0-Real Spectral Triple)......Page 178
5.4 Proposition......Page 179
5.5 Matrix-Form of   D (A); J  D (A)J; ad(G) and D......Page 180
6 The S0-Real Spectral Triple of the Full Standard Model......Page 182
6.1 De nitions......Page 183
6.3 Matrix Form......Page 184
6.4 Matrix-Form......Page 185
6.5 Matrices......Page 186
6.6 Matrix Form of the Covariant Dirac Operator......Page 187
6.7 Conversion into Classical Objects......Page 189
6.8 Canonical Decomposition......Page 191
7 The Spectral Action and Its Heat-Kernel Asymptotic Expansion......Page 195
7.1 De nition......Page 196
7.2 Asymptotic Expansion of the Spectral Action......Page 197
7.3 Remark......Page 198
7.4 Spectral Action Computation Program......Page 199
7.5 Computation of Fiber-Traces......Page 200
7.6 Gathering the Pieces......Page 206
8 Tree-Approximation Results......Page 207
9.1 Labeling the Basis......Page 210
9.2 Euclidean Fermionic Action:......Page 211
10 Does the Inner Spectral Triple of the Full Standard Model Proceed from a Quantum Group?......Page 213
10.1 The Hopf Algebra H1 and Its Regular Representation......Page 214
A Heat-Kernel Expansion......Page 216
A.2 Properties of the ej(x; P)......Page 217
A.4 Generalized Laplacians......Page 218
B.1 Definitions......Page 219
B.2 Remarks......Page 220
B.3 Proposition-De nition......Page 221
B.4 Lemma......Page 222
B.5 Proposition (Local Description of [B.3])......Page 223
C.1......Page 224
C.3......Page 225
D.1 Definition......Page 226
D.2 Lemma......Page 227
Bibliography......Page 228
1 Introduction and Motivation......Page 232
2 Ultraviolet Regularization......Page 236
3 Finite-Dimensional Algebras......Page 247
4 Di erential Calculi......Page 249
5 Yang-Mills Connections......Page 254
6 Metrics and Linear Connections......Page 256
7 In nite-Dimensional Models......Page 260
8 Gravity......Page 266
References......Page 269
1.1 Introduction......Page 275
1.2 The Laughlin Argument......Page 277
1.3 Thouless, Kohomoto, Nightingale, and den Nijs......Page 278
1.4 J. Avron, R. Seiler, Q. Niu, D. J. Thouless......Page 280
1.5 J. Bellissard, H. Schulz Baldes, A. Connes......Page 282
1.6 J. Fr¨ohlich, Q. Niu, X. G. Wen, A. Zee......Page 283
2.1 The Adiabatic Setup......Page 285
2.2 Kato's Equation......Page 286
2.3 The Adiabatic Theorem......Page 288
2.4 Adiabatic Curvature and Applications......Page 290
3.1 The QHE for Interacting Fermion Systems......Page 291
3.2 Fluctuations and Quillen's Formula......Page 293
3.3 Quantum Viscosity......Page 296
4.1 The Algebra of Two Projectors......Page 299
4.2 First Order Calculus on A := R (1......Page 300
4.3 The Index of a Pair of Projections......Page 302
4.4 Index Approach to the QHE......Page 303
4.5 Edge vs. Bulk......Page 305
References......Page 309
Introduction......Page 311
1.1 A Calculus Based on an Algebra......Page 312
1.2 Field Equations in a Purely Algebraic Context......Page 317
1.3 Gauge Theories in a Purely Algebraic Context......Page 319
1.4 q-Fourier Transformations......Page 326
1.5 Representations......Page 330
1.6 The De nite Integral and the Hilbert Space L 2 q......Page 336
1.7 Variational Principle......Page 340
1.8 The Hilbert Space L......Page 343
1.9 Gauge Theories on the Factor Spaces......Page 348
References......Page 350
2.1 SLq(2), Quantum Groups and the R-Matrix......Page 351
2.2 Quantum Planes......Page 355
2.3 Quantum Derivatives......Page 360
2.4 Conjugation......Page 366
2.5 q-Deformed Heisenberg Algebra......Page 369
2.6 The q-Deformed Lie Algebra slq(2)......Page 371
2.7 q-Deformed Euclidean Space in Three Dimensions......Page 376
References......Page 381
Abstracts of the Seminars......Page 383