Geometric Methods in Physics XXXVI: Workshop and Summer School, Białowieża, Poland, 2017

Contents......Page 6
Preface......Page 10
Part I Quantum Mechanics and Mathematics Twareque Ali in Memoriam......Page 12
 In Memory of S. Twareque Ali......Page 13
  1. Remembering Twareque......Page 14
  2. A short scientific biography......Page 15
  3. Concluding thoughts......Page 17
  References......Page 18
  1. Introduction......Page 19
  2. Noncommutative two-dimensional harmonic oscillator with linear terms......Page 20
  3. Bi-coherent states......Page 23
   3.1. A general theorem......Page 24
   3.2. Back to Swanson......Page 26
   3.3. What if V is complex......Page 27
  References......Page 28
  1. Introduction......Page 30
  2. Joint Measurability......Page 32
  3. Universal Markov kernel......Page 35
  References......Page 37
  1. Introduction......Page 39
  2. Balanced metric on the Siegel–Jacobi ball via coherent states......Page 40
  Geometric characterization of DJ n and Berezin quantization......Page 42
  References......Page 43
 1D & 2D Covariant Affine Integral Quantizations......Page 46
   2.1. The group framework......Page 47
   2.2. Affine quantization......Page 48
   2.3. Some formulae......Page 49
   2.4. Semi-classical portraits......Page 50
  3. Affine Wigner integral quantization......Page 51
  References......Page 52
  1. Introduction......Page 53
  2. Diffeomorphism groups in Galilean quantum theory......Page 54
  3. Hierarchies of representations......Page 55
  4. The diffeomorphism group and the free relativistic neutral scalar Bose field......Page 56
   5.1. Motivation......Page 57
   5.2. Anticipated steps......Page 58
   5.3. Constructing the relativistic free neutral scalar field......Page 59
  6. Discussion and concluding remarks......Page 60
  References......Page 61
Part II Noncommutative Geometry......Page 63
  1. Motivation and introduction......Page 64
  2. Generalized down-up and Weyl algebras......Page 66
  3. Skew derivations on generalized down-up algebras......Page 67
  References......Page 71
  1. Introduction......Page 73
  2. Introduction......Page 74
   2.1. Canonical spectral triple......Page 75
   2.2. Hodge–deRham spectral triple......Page 77
  3. Noncommutative formulation of the Standard Model: νS.M.......Page 79
   3.1. Properties of νS.M.......Page 81
   3.2. The geometric nature of Hf......Page 82
  References......Page 85
  1. Introduction......Page 87
  2. Noncommutative Minkowski phase space......Page 89
   2.1. NC Hamilton function and NC symplectic form......Page 90
   2.2. NC Poisson bracket and NC Hamiltonian vector field......Page 91
   3.1. NC Hamilton–Jacobi equation and generating function......Page 92
   3.2.(1,1)-tensor field T as recursion operator......Page 93
  4. Concluding remarks......Page 95
  References......Page 96
  1. Introduction......Page 98
   2.1. Conformally rescaled spectral triples......Page 99
   3.1. How to decompactify a circle......Page 100
   3.2. The “fuzzy” circle......Page 101
   3.3. Conformally rescaled fuzzy circle......Page 102
  References......Page 103
  1. Introduction......Page 105
  2. Noncommutative Toeplitz torus......Page 106
  3. Spectral triple on the noncommutative Toeplitz torus......Page 107
  References......Page 111
Part III Quantization......Page 112
  2. Model......Page 113
  3. Partition function......Page 115
  References......Page 118
 Quantization of Mathematical Theory of Non-Smooth Strings......Page 119
   1.2. Algebra of observables Ad......Page 120
   1.4. The action of the diffeomorphism group of the circle on Ωd......Page 121
   2.2. Fock space associated with Vd......Page 122
   2.3. Heisenberg representation......Page 123
   2.4. Quantization of the algebra Ad......Page 124
   3.2. Quasisymmetric homeomorphisms......Page 125
   3.3. Nag–Sullivan theorem......Page 126
   3.4. Connes quantization......Page 127
   3.5. Quantization of the Sobolev space Vd......Page 128
  References......Page 129
  1. Presentation......Page 130
   1.2. Some motivating quotes......Page 131
   2.1. Deformation quantization and avatars......Page 132
   2.2. Symmetries of elementary particles......Page 133
   2.3. Combining both, and perspectives......Page 134
  Acknowledgment......Page 135
  References......Page 136
  1. Introduction......Page 138
  2. Classical statistical mechanics......Page 139
  3. Physical background of formal series calculus......Page 140
  4. States over the commutative ring (C∞[λ−1, λ]](M), •)......Page 141
  5. States over the algebra (C∞[λ−1, λ]](M), ∗)......Page 142
  References......Page 144
  1. General theory......Page 146
  2. The case of algebraic varieties......Page 148
  3. “Desingularizing” the definition......Page 149
  References......Page 153
   1.1. Definition of star products......Page 154
   1.2. Equivalence, Star product algebra bundle and flat connection......Page 155
  2. Star exponential......Page 156
   2.1. Star exponential of linear and quadratic polynomials......Page 157
    2.2.1. Linear case.......Page 159
   2.3. Star exponentials of quadratic polynomials......Page 161
    2.3.2. Vacuum.......Page 162
  References......Page 163
Part IV Integrable Systems......Page 164
  1. Introduction. Haantjes tensors generalize recursion operators......Page 165
   2.3. The Nijenhuis torsion......Page 166
   2.5. The Frölicher–Nijenhuis bracket......Page 167
   3.3. First citations of Haantjes’s article......Page 168
   4.2. Lie algebroids......Page 169
   4.5. Higher Haantjes torsions......Page 170
   5.1. From Nijenhuis to Haantjes manifolds......Page 171
   5.3. Magri–Lenard complexes generalize Lenard chains......Page 172
   5.4. A Magri–Lenard complex on R3......Page 173
  6. WDVV equations and Magri–Lenard complexes......Page 175
  Acknowledgment......Page 176
  References......Page 177
  1. Introduction......Page 179
   1.1. Motivations......Page 180
  3. Review of formally real Jordan algebras......Page 181
   3.2. The structure algebra......Page 182
  4. The universal Kepler problems......Page 183
   5.2. Isotropic oscillator in dimension n......Page 184
  References......Page 185
  1. Introduction and main results......Page 186
  2. DKN equation......Page 189
  3. Proof of Theorem 1......Page 191
  References......Page 192
Part V Differential Geometry and Physics......Page 193
  1. Introduction......Page 194
  2. The dressing field method in a nutshell......Page 196
   3.2.SU(2)-erasing without symmetry breaking......Page 197
   4.1. The usual bottom-up construction......Page 198
   4.2. A top-down construction via dressing field method......Page 199
  References......Page 200
  1. Axiomatics of two-dimensional oriented open-closed TFTs......Page 201
  2. B-type open-closed Landau–Ginzburg theories......Page 203
   2.1. The off-shell bulk algebra......Page 204
   2.3. The full TFT data......Page 205
   3.1. An analytic model for the on-shell bulk algebra......Page 206
  Acknowledgment......Page 207
  References......Page 208
  1. Introduction and basic notions......Page 209
  2. The notion of the Dirac type operator determined by a skew-symmetric tensor......Page 211
  3. The Weyl module of the bundle of symmetric tensors with values in a vector bundle and the Dirac operator on symmetric tensors......Page 212
  4. The Dirac symmetric operator, the symmetric derivative and coderivative......Page 213
  References......Page 215
  1. Introduction......Page 217
   2.2. Cartan integral invariants......Page 218
   3.1. Stationary case......Page 219
   3.2. General, non-stationary case......Page 220
  References......Page 221
  1. Introduction......Page 223
  2. Real Lipschitz structures and their relation to real pinor bundles......Page 224
  3. Applications to string theory and supergravity......Page 227
  References......Page 228
  1. Introduction......Page 230
   2.1. Conic Minkowski norm on a Banach space......Page 231
   2.2. Conic sub-Hilbert–Finsler structure......Page 233
   3.2. Energy, length and semi-distance......Page 234
   3.3. Hamiltonian characterization of normal extremal......Page 236
  References......Page 237
  1. Introduction......Page 238
  2. Preliminaries......Page 239
  3. R-quadratic Spherically Symmetric Finsler Metrics in Rn......Page 240
  4. Ricci-quadratic spherically symmetric Finsler metrics in Rn......Page 242
  References......Page 243
Part VI Topics in Spectral Theory......Page 244
  1. Introduction......Page 245
  2. Toy model of renormalization group......Page 246
  3. Homogeneous Schrödinger operators......Page 248
  4. Almost homogeneous Schrödinger operators......Page 252
  References......Page 255
  1. Introduction......Page 257
  2. General scattering systems in one dimension......Page 259
  3. Generalized unitarity relation......Page 261
  Acknowledgment......Page 263
  References......Page 264
   2.1. Polyhedral surfaces......Page 265
   2.2. Definitions of Laplacians......Page 266
  3. Spaces of harmonic functions......Page 267
  4. Trace formulas......Page 268
  6. Lagrangian manifolds, corresponding to localized solutions......Page 269
  References......Page 270
Part VII Representation Theory......Page 271
  1. Introduction......Page 272
   Representation theory......Page 273
  3. Pro-Lie groups and their Lie algebras......Page 274
  4. Main results......Page 276
  References......Page 278
  1. Branching problems – Stages A to C......Page 279
  2. Preliminaries: smooth representations......Page 281
  3. Multiplicities in symmetry breaking......Page 282
  4. Conformally covariant SBOs......Page 284
  5. Classification theory of conformally covariant differential SBOs......Page 287
  6. Classification theory: nonlocal conformally covariant SBOs......Page 290
  7. Application to periods and automorphic form theory......Page 293
  References......Page 296
  1. Introduction......Page 299
  2. Fock representation for a generalized statistics......Page 303
   3.1. One-dimensional lattice......Page 305
   3.2. Two-dimensional lattice......Page 307
   4.1. An ACR algebra......Page 310
   4.2. Non-Fock representations......Page 313
  References......Page 316
Part VIII Special Topics......Page 318
  1. Introduction......Page 319
  2. Friedrichs model......Page 322
  3. A no-go-theorem......Page 324
  4. A result for scattering systems with poles of the scattering matrix in the upper half-plane......Page 325
  References......Page 328
  1. Introduction......Page 329
  2. Root and weigh lattices of An......Page 330
  3. Construction of the graphene from the Lie algebra A2......Page 331
  4. Method of dynamical generation......Page 333
  References......Page 334
  1. Introduction......Page 335
  2. Eight kinds of polynomials of C2......Page 337
  3. Weight functions and discrete orthogonality......Page 340
  Acknowledgment......Page 341
  References......Page 342
  1. Counting of branched covers......Page 343
  2. Random matrices. Complex Ginibre ensemble......Page 349
   Appendix A. Partitions and Schur functions......Page 353
   Appendix B. Matrix integrals as generating functions of Hurwitz numbers from [48, 49]......Page 354
  Acknowledgment......Page 357
  References......Page 358
Part IX Extended Abstracts of the Lectures at “School on Geometry and Physics”......Page 362
  2. Poincaré integral invariants......Page 363
   2.1. Stationary Euler equation......Page 364
   2.3. Why the form of Eq. (10) is so convenient......Page 365
  3. Cartan integral invariants......Page 366
   3.2. Helmholtz theorem on frozen vortex lines – non-stationary case......Page 367
  References......Page 368
  2. Hilbert C*-modules and adjointable maps......Page 369
  3. C*-correspondences......Page 371
  4. The Morita–Rieffel equivalence......Page 373
  References......Page 374
  1. Abstract Plancherel theorem for groups......Page 375
  2. An example. The group GL(2,R)......Page 376
  4. Operational calculus for GL(2, R), see [33], 2017......Page 377
  5. Difference operators in imaginary direction and classical integral transforms......Page 379
  6. A general problem about overalgebras......Page 380
  8. Example: separation of series for the one-sheet hyperboloid......Page 381
  9. Splitting off the complementary series, see [35]......Page 383
  References......Page 385
  2. What is noncommutative geometry?......Page 388
  3. From spaces to algebras (and back)......Page 389
   4.1. Differential Calculi......Page 390
  5. Spectral triples and how to use them......Page 391
   5.2. Differential forms and fluctuations......Page 392
   5.5. Towards (noncommutative) physics......Page 393
   5.7. Where can you learn more?......Page 394
  1. Motivations......Page 395
  2. The universal Teichm¨uller space......Page 396
  3. The restricted Siegel disc......Page 399
  4. The period mapping......Page 401
  References......Page 402
   1.1. Hybrid spaces......Page 403
   2.1. Hybrid spaces......Page 404
  3. Spaces of harmonic functions......Page 405
   4.1. The counting function for geodesics......Page 406
  6. Case of the polynomial growth of m(t)......Page 407
  References......Page 408