Geometric and Topological Methods for Quantum Field Theory

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
List of contributors......Page 11
Introduction......Page 13
1.1 Geometric topology: a brief history......Page 15
1.1.1 Examples......Page 16
1.1.2 Invariants......Page 18
1.1.3 High and low dimensions......Page 19
1.1.4 Links, Reidemeister moves, and Kirby's theorem......Page 21
1.2 Homology theories......Page 23
1.3 Axioms derived from QFT......Page 27
1.3.1 Formal properties of functional integrals......Page 28
1.3.2 Axioms for a TQFT......Page 34
1.4.1 Outline of the construction and properties......Page 36
1.4.2 Some results proven using Donaldson-Floer theory......Page 43
1.5.1 Spincccc structures......Page 45
1.5.2 Outline of the construction of SW invariants for closed 4-manifolds......Page 46
1.5.3 Some results obtained from Seiberg-Witten theory......Page 48
1.6.1 Lagrangian Floer homology......Page 50
1.6.2 Heegaard splittings of 3-manifolds......Page 52
1.6.3 Heegaard-Floer homology......Page 53
1.7 TQFTs in dimension 2+1......Page 56
1.7.1 Modular tensor categories......Page 57
1.7.2 Construction of a TQFT from a modular tensor category......Page 60
Bibliography......Page 64
Abstract......Page 68
2.1 Linear differential equations and D-modules......Page 69
2.2 The quantum differential equations......Page 75
2.3 A D-module construction of integrable systems......Page 83
2.4.1 The WDVV equation and reconstruction of big quantum cohomology......Page 87
2.4.2 Crepant resolutions......Page 90
2.4.3 Harmonic maps and mirror symmetry......Page 92
2.5 Conclusion......Page 96
Bibliography......Page 97
3.1 Introduction......Page 100
3.2.1 Definitions and basic examples of groupoids......Page 103
3.2.2.2 Morita equivalence......Page 105
3.2.4 Groupoids associated to a foliation......Page 107
3.2.5 The noncommutative tangent space of a conical pseudomanifold......Page 109
3.2.6 Lie theory for smooth groupoids......Page 110
3.2.7.1 The tangent groupoid......Page 112
3.2.7.2 The Thom groupoid......Page 113
3.2.8 Haar systems......Page 114
3.3.1 C*-algebras - Basic definitions......Page 115
3.3.1.1 Enveloping algebra......Page 116
3.3.2 The reduced and maximal C*-algebras of a groupoid......Page 117
3.4.1 Historical comments......Page 120
3.4.2 Abstract properties of KK(A,B)......Page 122
3.5.1 Basic definitions and examples......Page 125
3.5.2.1 Adjoints......Page 129
3.5.2.2 Orthocompletion......Page 132
3.5.2.4 Polar decompositions......Page 134
3.5.2.5 Compact homomorphisms......Page 135
3.5.3 Generalized Fredholm operators......Page 137
3.5.4.1 Inner tensor products......Page 138
3.5.4.3 Connections......Page 139
3.6.1 Kasparov modules and homotopies......Page 140
3.6.3.2 Atiyah’s Ell......Page 142
3.6.3.4 (Quasi) Self-adjoint representatives......Page 143
3.6.3.6 Relationship with ordinary K-theory......Page 144
3.6.4 Ungraded Kasparov modules and KK1......Page 145
3.6.5 The Kasparov product......Page 147
3.6.6 Equivalence and duality in KK-theory......Page 149
3.6.6.1 Bott periodicity......Page 150
3.6.6.2 Self-duality of C0(R)......Page 152
3.6.6.4 C0(R) and C1.......Page 153
3.6.7.2 Maps between K-theory groups......Page 154
3.6.7.3 Kasparov elements constructed from homomorphisms......Page 155
3.7 Introduction to pseudodifferential operators on groupoids......Page 156
3.8.1 The KK-element associated to a deformation groupoid......Page 162
3.8.2 The analytical index......Page 163
3.8.3 The topological index......Page 164
3.9 The case of pseudomanifolds with isolated singularities......Page 166
3.9.2 The Poincare duality......Page 167
3.9.3.1 Thom isomorphism......Page 168
3.9.3.3 Bott element......Page 169
Bibliography......Page 170
4.1 Algebras of representative functions......Page 173
4.1.2 Coproduct......Page 174
4.1.4 Antipode......Page 175
4.1.7 Representative functions......Page 176
4.2.1 Complex affine plane......Page 177
4.2.2 Real affine plane......Page 178
4.2.4 Complex general linear group......Page 179
4.2.5 Simple unitary group......Page 180
4.2.7 Exercise: Euclidean group......Page 181
4.2.8 Group of invertible formal series......Page 182
4.2.9 Group of formal diffeomorphisms......Page 183
4.3.2 Real subgroups......Page 185
4.3.4 Comparison of SL(2,C), SL(2,R) and SU(2)......Page 186
4.3.6 Algebraic and proalgebraic groups......Page 187
4.4.3 Euler-Lagrange equation......Page 188
4.4.4 Free and interacting Lagrangian......Page 189
4.4.5 Free fields......Page 190
4.4.6 Self-interacting fields......Page 191
4.5.2 Green functions through path integrals......Page 192
4.5.4 Dyson-Schwinger equation......Page 193
4.5.5 Connected Green functions......Page 194
4.5.6 Self-interacting fields......Page 195
4.5.8 Conclusion......Page 196
4.6.1 Feynman notation......Page 197
4.6.3 Perturbative expansion on trees......Page 198
4.6.5 Conclusion......Page 199
4.7.3 Dyson-Schwinger equations......Page 200
4.7.4 Perturbative expansion on graphs......Page 201
4.7.6 Feynman rules......Page 202
4.7.9 Conclusion......Page 203
4.8.1 Momentum coordinates......Page 204
4.8.3 One-particle irreducible graphs......Page 205
4.8.4 Proper or 1PI Green functions......Page 206
4.8.5 Conclusion......Page 207
4.9.1 Problem of divergent integrals: ultraviolet and infrared divergences......Page 208
4.9.2 Renormalized amplitudes, normalization conditions and renormalizable theories......Page 209
4.9.3 Power counting: classification of one loop divergences......Page 210
4.9.4.1 Regularization......Page 211
4.9.5 Renormalization of a simple loop: Bogoliubov's subtraction scheme......Page 212
4.9.6 Local counterterms......Page 213
4.9.7.2......Page 214
4.9.7.3......Page 215
4.9.9 Renormalization of many loops: BPHZ algorithm......Page 216
4.9.10 Recursive definition of the counterterms......Page 218
4.9.11.1......Page 219
4.9.11.2......Page 220
4.10.1 Bare and renormalized Lagrangian......Page 221
4.10.2 Renormalization factors......Page 222
4.10.3 Bare and effective parameters......Page 223
4.10.5 Renormalization and semidirect product of series......Page 224
4.11 Connes-Kreimer Hopf algebra of Feynman graphsand diffeographisms......Page 225
4.11.2 Hopf algebra of Feynman graphs......Page 226
4.11.3 Group of diffeographisms and renormalization......Page 227
4.11.4 Diffeographisms and diffeomorphisms......Page 228
4.11.5 Diffeographisms as generalized series......Page 229
4.11.7 Groups of combinatorial series......Page 230
Bibliography......Page 232
5.1.1 A historical note......Page 234
5.1.2 Utiyama's analysis, first part......Page 235
5.1.3 Final touches to the Lagrangian......Page 239
5.1.4 The electromagnetic field......Page 241
5.1.5 The original Yang-Mills field......Page 242
5.2.1 What is wrong with the Proca field?......Page 243
5.2.2 What escaped through the net......Page 245
5.2.3 The Stuckelberg field and Utiyama's test......Page 246
5.2.4 The Stuckelberg formalism for non-abelian Yang-Mills fields......Page 248
5.2.5 Gauge-fixing and the Stuckelberg Lagrangian......Page 249
5.2.6 The ghosts we summoned up?......Page 251
5.3.1 On the need for BRS invariance......Page 253
5.3.2 Ghosts as free quantum fields......Page 255
5.3.3 Mathematical structure of BRS theories......Page 257
5.3.4 BRS theory for massive spin-one fields......Page 260
5.3.5 The ghostly Krein operator......Page 262
Bibliography......Page 265
6.1 Introduction......Page 267
6.2 Strings: a geometric dynamical object......Page 268
6.2.1 The Polyakov string: more is the same......Page 271
6.3 Solving the Polyakov string......Page 273
6.3.1 The Euclidean setup......Page 274
6.3.2 Going back to real time: Fourier series and quantization......Page 275
6.3.3 Massless particles and superstrings......Page 277
6.4 Open strings and a stack of branes: setting upthe AdS/CFT conjecture......Page 280
6.4.1 The low-energy limit......Page 283
6.4.2 A different point of view: gravity......Page 285
6.5 Making sense of it all: observables in CFT and gravity......Page 289
6.6 The operator state correspondence, the superconformal group and unitary representations......Page 291
6.7 Matching of BPS representations......Page 294
6.8 Recent developments......Page 298
Bibliography......Page 299
7.1 Introduction......Page 302
7.2.1 The basic construction for scalar fields......Page 304
7.2.1.1......Page 306
7.2.1.3......Page 307
7.2.2 Theory space......Page 308
7.2.3 Nonperturbative approximations through truncations......Page 310
7.3 The effective average action for gravity......Page 312
7.4 Truncated flow equations......Page 319
7.4.2......Page 325
7.4.3......Page 326
7.5 Asymptotic Safety......Page 327
7.6.1 The phase portrait of the Einstein-Hilbert truncation......Page 330
7.6.2 Testing the Einstein-Hilbert truncation......Page 331
7.6.3.3 Stability......Page 332
7.6.3.5 Higher and lower dimensions......Page 333
7.6.3.7 Eigenvalues and -vectors (R2)......Page 335
7.6.3.8 Scheme dependence (R2)......Page 337
7.7 Discussion and conclusion......Page 338
Bibliography......Page 341
Introduction......Page 344
8.1 Cobordism of manifolds......Page 345
8.1.2 Characteristic numbers......Page 346
8.1.3 Oriented cobordism......Page 348
8.2.1 Charts......Page 350
8.2.2 Orbibundles......Page 351
8.3 Cobordism of orbifolds......Page 353
8.3.2 The unoriented case......Page 354
8.3.3 The oriented case......Page 355
Bibliography......Page 357
9.1 Introduction......Page 358
9.2 Canonical group quantization......Page 361
9.3.1 The classical problem......Page 366
9.3.2 The quantum problem......Page 368
9.3.3 Local description of Ln......Page 370
9.3.4 Construction of the angular momentum operators......Page 371
9.4.1 Configuration space......Page 374
9.4.2 Construction of the angular momentum operators......Page 375
9.5 Conclusions......Page 378
Bibliography......Page 379
10.1 Introduction......Page 382
10.2 Lie algebroids......Page 383
10.3 Bidifferential calculi, PN structures and Lie algebroids......Page 385
10.4 Examples......Page 391
10.5 Final Comments......Page 392
Bibliography......Page 393
Introduction......Page 395
11.1 Log-polyhomogeneous operators......Page 396
11.2 Holomorphic families of log-polyhomogeneous operators......Page 398
11.3 Symmetrized logarithms......Page 400
11.4 The canonical trace extended to odd-class operators in odd dimension......Page 401
11.5 The symmetrized canonical determinant......Page 404
Bibliography......Page 407
12.1 Introduction......Page 408
12.2 Maximal flag manifolds......Page 409
12.3 Cosymplectic metrics on F(n)......Page 412
Bibliography......Page 417
13.1 Introduction......Page 419
13.2.1 The C*-algebra A......Page 420
13.2.2 Smooth elements......Page 421
13.2.3 Vector bundles and K-theory......Page 422
13.2.4 Morphisms of noncommutative tori......Page 423
13.3 Heisenberg groups and their representations......Page 425
13.3.1 Real Heisenberg groups......Page 427
13.3.2 Heis(( Z/cZ)2)......Page 429
13.4 Heisenberg modules over noncommutative tori with real multiplication......Page 430
13.5 Some rings associated to noncommutative tori with real multiplication......Page 432
Bibliography......Page 434