Quantum theory and symmetries with Lie theory... Vol. 1

Preface of Volume 1......Page 6
 Acknowledgements......Page 7
Contents......Page 8
List of Participants......Page 11
 Summary......Page 15
Plenary Talks......Page 16
 1 Introduction......Page 17
  1.1 Conformal Representations I(λ) and J(ν) Associated with Pseudo-Riemannian Manifolds mathbbRp,q supsetmathbbRp-1,q......Page 18
  1.3 Construction of Symmetry Breaking Operators......Page 19
  1.4 Image of the Symmetry Breaking Operators mathbbAλ,ν......Page 21
  2.1 Finiteness Criterion for Multiplicities in Branching  of Representations......Page 25
  2.2 Program ABC for Branching......Page 27
  3.1 Subsets / /, midmidmid, \\\\ and mathbbX in mathbbC2......Page 28
  3.2 Description of Submodules of the Principal Series J(ν)......Page 29
  3.3 Main Theorem: Image of mathbbAλ,ν......Page 31
  3.4 Restatement of Theorem 15......Page 32
  4.1 Socle Filtration of the Target Space J(ν)......Page 33
  4.2 Reduction of the Parameter Set for (λ,ν)......Page 34
  4.3 Description of the Image of Symmetry Breaking Operators (p Even, q Even)......Page 35
 5 Graphic Description of the Image of the Regular Symmetry Breaking Operator: Case p Odd (ge3) and q Even......Page 37
  5.1 Socle Filtration of the Target Space J(ν)......Page 38
  5.3 Description of the Image of Symmetry Breaking Operators (p Odd 3, q Even)......Page 39
  6.2 Reduction of the Parameter Set for (λ,ν)......Page 42
  6.3 Description of the Image of Symmetry Breaking Operators (p Even, q Odd)......Page 43
  7.1 Socle Filtration of the Target Space J(ν)......Page 45
  7.3 Description of the Image of the Symmetry Breaking Operators mathbbAλ,ν (p Odd ge3, q Odd)......Page 46
 References......Page 48
Trails for Minuscule Modules and Dual Kashiwara Functions for the B(infty) Crystal......Page 50
  1.2 Describing BJ(infty)......Page 51
  1.4 Trails......Page 52
  1.5 Functions......Page 53
  1.7 S-graphs......Page 54
  2.2 Adjoining Faces to Trails......Page 55
  2.4 The Driving Function......Page 56
  2.6 The Final Function......Page 57
  2.7 The S-set of a Totally Degenerate S-graph......Page 58
  3.1 Announcement of Main Thorem......Page 59
  3.4 Moving es Factors......Page 60
  3.5 Sequences for Fixed s inI......Page 61
  3.7 Strategy......Page 62
  3.8 Removing Faces......Page 63
  3.10 Conclusion of Proof of Main Theorem......Page 64
 4 Index of Notation......Page 65
 References......Page 66
From Euler\'s Play with Infinite Series  to the Anomalous Magnetic Moment......Page 67
  1.1 The Basel Problem......Page 68
  1.2 Memorable Mathematical Developments......Page 70
  2.1 QED and the Intrinsic Magnetic Moment of the Electron (1947–1949)......Page 72
  2.2 Fourth and Sixth Order Corrections (1949–1996)......Page 73
  2.3 Eight\'s Order and Beyond (1996–2017)......Page 76
 3 Periods in Quantum Field Theory......Page 77
  3.1 The Ring of Periods......Page 78
  3.2 A Galois Group for Periods?......Page 80
 4 Outlook......Page 82
 References......Page 83
 1 Introduction......Page 86
  2.1 Some Arithmetic......Page 87
  2.2 Relating the Arithmetic to Geometry......Page 88
  2.3 Differences......Page 90
  2.4 Interpolations Galore......Page 91
 3 The Wallis Formula from Quantum Mechanics......Page 97
 References......Page 100
 1 Introduction......Page 101
 2 Variation of Curvature Tensors......Page 103
 3 Equations of Motion......Page 109
 4 Second Variation of the Action......Page 111
  5.2 Perturbations......Page 117
 6 Concluding Remarks......Page 119
 References......Page 120
 1 Ortho-Conformal and Meta-Conformal Invariances......Page 122
  2.1 Physical Background......Page 127
  2.2 Impossibility of a Local Meta-Conformal Invariance......Page 130
  2.4 Non-local Meta-Conformal Generators......Page 131
  2.5 Ward Identities......Page 133
  2.6 Co-variant Two-Time Correlator and Response......Page 134
 3 Meta-Conformal Algebra in d=2 Spatial Dimensions......Page 136
  3.1 The Case p=-1......Page 137
  3.2 The Case p=1/3......Page 140
 References......Page 143
 1 Introduction......Page 145
 2 Polarization Tensors for the Field of Arbitrary Spin......Page 146
 3 Spin Projection Operator for Arbitrary Spin......Page 150
 References......Page 155
 1 Introduction......Page 157
 2 Quantum Affine Superalgebra Uq(sl\"0362sl(M|N))......Page 158
  3.1 Boson......Page 159
  3.2 Bosonization......Page 161
  3.3 Wakimoto Realization......Page 164
 4 Screening Operator......Page 166
 5 Limit q to1......Page 167
 References......Page 171
 1 Introduction......Page 172
 2 Preliminaries and Definitions......Page 173
 3 Dimension, Superdimension and t-Dimension......Page 174
 4 Superdimensions for mathfrakosp(2m+1|2n) and mathfrakosp(2m|2n)......Page 177
 5 Characters of ``fork\'\' Representations for mathfrakso(2m) and mathfrakosp(2m|2n)......Page 180
 References......Page 183
 1 Introduction......Page 184
  2.1 CSDR of a D-dimensional Theory......Page 187
  2.2 The 4-dimensional Effective Action......Page 189
  3.1 The Fuzzy Sphere......Page 191
  3.2 Gauge Theory on Fuzzy Sphere......Page 194
  4.1 Ordinary Fuzzy Dimensional Reduction......Page 196
  4.2 Fuzzy CSDR......Page 197
 5 Orbifolds and Fuzzy Extra Dimensions......Page 198
  5.1 mathcalN=4 SYM Field Theory and mathbbZ3 Orbifolds......Page 199
  5.2 Dynamical Generation of Twisted Fuzzy Spheres......Page 200
  5.3 Chiral Models After the Fuzzy Orbifold Projection - The SU(3)ctimesSU(3)LtimesSU(3)R Model......Page 203
  6.1 4-Dimensional Gravity as a Gauge Theory......Page 205
  6.2 3-Dimensional Gravity as a Gauge Theory......Page 206
  6.3 3-Dimensional Gravity as a Gauge Theory on Non-commutative Spaces......Page 207
 7 Conclusions......Page 208
 References......Page 209
Representation Theory......Page 213
 1 Introduction......Page 214
 2 Construction of a mathcalT-Operator......Page 215
  2.1 DIM Algebra and Representations......Page 216
  2.2 From (p,q)-Brane Webs to Representations Webs......Page 217
  2.3 Integrability......Page 219
 3 Defining qq-Characters......Page 220
 4 Discussion......Page 221
 References......Page 222
 1 SO(2,q), mathfrakso(2,q) and Subgroups of SO0(2,q)......Page 224
 3 Geometrical Results......Page 228
 4 Parallelizations of Induced Bundles and Representations of SO0(2, q)......Page 230
 5 A Generalization of the Oscillator Parallelization  for SO0(2,q) (q even, q>3)......Page 232
 6 Results on Various Parallelizations for the Representations π(s,ε,σ)(SO0(2,q))......Page 233
 References......Page 234
 1 Introduction......Page 235
 2 Jordan Algebra and Its Structure Algebra......Page 236
 3 Tits–Kantor–Koecher Construction......Page 238
 4 Causal Automorphisms of Space-Time......Page 239
 5 Hydrogen Atom and Its Spectrum Generating Algebra......Page 242
 6 Causal Automorphisms Versus Spectrum  Generating Algebra......Page 246
 References......Page 248
 1 Introduction......Page 249
 2 Free Field Realization......Page 252
 3 Screening Operators for the Lattice VOA......Page 254
 References......Page 256
 1 Introduction......Page 258
 2 The S3 Dirac–Dunkl Operator......Page 259
 3 Symmetries of the Dirac–Dunkl Operator......Page 260
 References......Page 263
Integrability......Page 264
The Superintegrable Zernike System......Page 265
 2 The Classical Zernike System......Page 266
 3 The Quantum Zernike System......Page 268
 4 Interbasis Expansions......Page 271
 5 Algebraic Structure of Zernike\'s System......Page 273
 6 Conclusions......Page 274
 References......Page 275
 1 Introduction......Page 276
 2 Twisted Vertex Algebras Related to the Two Bosonizations of the CKP Hierarchy......Page 278
 3 Bicharacter Construction of Twisted Vertex Algebra Examples......Page 283
 4 Bicharacter Realizations of the Twisted Vertex Algebras Arising from the CKP Hierarchy......Page 287
 References......Page 292
 1 Introduction......Page 294
 2 The smKdV Hierarchy......Page 296
 3 Recursion Operator for smKdV Hierarchy......Page 300
 4 The Bäcklund Transformations for the smKdV Hierarchy......Page 303
 5 Recursion Operator for the Bäcklund Transformations......Page 306
 References......Page 309
 1 Introduction......Page 311
 2 Lie Symmetries for Eq.(1)......Page 312
 3 Symmetry Analysis for Equation with Time-Dependent Coefficients......Page 315
 4 Symmetry Analysis for Equation with x-Dependent Coefficients......Page 317
 References......Page 318
Entanglement......Page 319
Probing Anderson Localization  Using the Dynamics of a Qubit......Page 320
 1 Introduction......Page 321
 2 Two-Level System......Page 322
 3 TLS Coupled to Infinite Chain......Page 323
 4 Tripartite System (Ordered Case)......Page 324
 5 Tripartite System (Disordered Case)......Page 326
 6 Summary......Page 327
 References......Page 328
 1 Introduction......Page 329
 2 Rotation Groups......Page 330
 3 General Lie Group Representations......Page 331
 References......Page 333
 1 Introduction......Page 334
 2 Closed Strings in the AdS5timesS5 T-Dual pp-Wave: The Abelian Case......Page 335
 3 Closed Strings in the AdS5timesS5 T-Dual pp-Wave: The Non-abelian Case......Page 337
 4 Conclusion......Page 340
 References......Page 341
Quantum Groups and Related Structures......Page 342
A Unified Approach to Poisson–Hopf Deformations of Lie–Hamilton Systems Based on mathfraksl(2)......Page 343
 1 Introduction......Page 344
 2 Lie–Hamilton Systems and Poisson–Hopf Algebras......Page 345
 3 Poisson–Hopf Deformations of Lie–Hamilton Systems......Page 347
 4 The Non-standard Poisson–Hopf Algebra Deformation  of mathfraksl(2)......Page 350
 5 Poisson–Hopf Deformations of mathfraksl(2) Lie–Hamilton Systems......Page 352
 7 A Method to Construct Lie–Hamilton Systems......Page 357
 8 Concluding Remarks......Page 361
 References......Page 362
 1 Introduction......Page 363
 2 RTT Realization and Orthogonality......Page 364
 3 Universal R-Matrix......Page 369
 References......Page 371
 1 Main Results......Page 373
 2 Constructions......Page 375
 References......Page 377
Various Mathematical Results......Page 378
 1 Introduction......Page 379
 3 Bundles Associated with Arrangements......Page 380
 4 Freeness Versus Infinitely Extendability......Page 381
 5 The Stably Freeness of Arrangements......Page 382
 6 A ``Stable\'\' Terao Conjecture......Page 383
 References......Page 384
Multiplication of Distributions  and Nonperturbative Calculations  of Transition Probabilities......Page 386
 2 A Mathematical Context in Which One Can Multiply Distributions......Page 387
 3 A Brief Recall on the Calculations in the Canonical Hamiltonian Formalism for a Neutral Massive Self-interacting Boson Field......Page 389
 4 Interpretation of QFT in this Mathematical Context......Page 390
 5 Calculations of an Infinite Number of Oscillations......Page 392
 References......Page 393
 1 Introduction......Page 395
 2 Gauge Theory of Trade Networks......Page 397
  3.1 Posets and Incidence Algebras......Page 398
  3.2 Mobius Inversion......Page 399
  4.1 Markov and Gibbs Fields on Graphs......Page 400
 5 Outlook and Speculations......Page 402
 References......Page 403
 1 Introduction......Page 405
 2 Preliminaries......Page 406
 3 Method for Classifying Filiform Lie Superalgebras......Page 408
 4 Classification in Low Dimensions......Page 409
 References......Page 412
 1 Introduction......Page 413
 2 Preliminaries......Page 414
 3 Equivalence......Page 415
 4 Transitive Groups......Page 416
 5 Formal Sum of Realizations......Page 417
 6 Conclusions......Page 418
 References......Page 419