Partition Functions and Automorphic Forms

Preface to the Book Series Moscow Lectures......Page 6
Preface......Page 9
Contents......Page 13
Contents......Page 14
1 Elliptic Curves......Page 15
2 Supersingular Elliptic Curves......Page 17
3 Ogg's Observation......Page 18
4 Complex Elliptic Curves......Page 20
5 Monstrous Moonshine......Page 22
6 Elliptic Curves with Level......Page 24
7 Affine Lie Algebras......Page 27
8 Modular Forms......Page 30
9 Vertex Operators......Page 33
10 Conformal Field Theory......Page 37
11 The Monster Module......Page 39
12 Affinization......Page 43
13 Vertex Algebras......Page 44
14 Vertex Operator Algebras......Page 47
15 Orbifolds......Page 49
16 The Genus Zero Problem......Page 51
17 Super Structure......Page 55
18 Super Moonshine......Page 58
19 Superconformal Structure......Page 61
20 Conway Moonshine......Page 64
21 The Mathieu Groups......Page 66
22 K3 Surfaces......Page 69
23 Complex Elliptic Genera......Page 71
24 Mathieu Moonshine......Page 74
25 Conway Returns......Page 79
26 Forward from Here......Page 83
References......Page 88
Contents......Page 99
1 Automorphic Correction of Elliptic Genus......Page 100
1.1 Witten Genus......Page 101
1.2 Elliptic Genus of Calabi–Yau Manifolds......Page 102
1.3 Complex Curves, Surfaces and Threefolds......Page 107
1.4 When χ(M,E;τ,z) is Holomorphic?......Page 108
2 The Graded Algebra J0,*/2 Over Z......Page 110
3.1 Calabi–Yau Fivefolds and the Euler Number Modulo 24......Page 119
3.2 Exact Formulae for MWG for Dimension d=6, 7, 9 and Rank r3.3 Relations Between -Genera......Page 124
3.4 Vector Bundle of Rank 2......Page 127
4 A2(2)-Genus......Page 129
References......Page 130
1 Introduction......Page 132
2 Instanton Partition Functions: Basics and Calculus......Page 134
3 Instantons in 5d QFTs......Page 143
3.1 5d Instantons for 5d SCFTs......Page 144
3.2 5d Instantons for 6d SCFTs on S1......Page 149
4 Instanton Strings in 6d QFTs......Page 156
4.1 Strings of 6d N=(2,0) Theories......Page 161
4.2 Strings of 6d N=(1,0) Theories......Page 163
5 Superconformal Observables......Page 167
5.1 6d (2,0) Index and its Physics......Page 168
5.2 5d Index and its Physics......Page 181
6 Conclusion......Page 183
References......Page 184
Contents......Page 189
1 Introduction......Page 190
2.1 Basic Definitions......Page 191
2.2 Maximal Compact Subalgebras......Page 194
3 Spin Representations......Page 198
3.1 The Simply-Laced Case......Page 200
3.1.1 AEd and K(AEd) (d>3)......Page 201
3.1.2 E10 and K(E10)......Page 205
3.1.3 DE10 and K(DE10)......Page 206
3.2.1 AE3 and K(AE3)......Page 208
3.2.2 G2++ and K(G2++)......Page 210
3.2.3 BE10 and K(BE10)......Page 211
4.1 Spin-12 Quotients in AEd......Page 213
4.2 Quotient Algebras for K(E10)......Page 215
4.2.1 Spin-32......Page 216
4.2.2 Spin-52 and Spin-72......Page 217
4.3 General Remarks on the Quotients......Page 218
5 K(E10) and Standard Model Fermions......Page 221
References......Page 223
Contents......Page 226
1 Introduction......Page 227
2 Three-Manifolds and the Categorification of WRT Invariants......Page 228
2.1 Fivebranes on Three-Manifolds and Categorification of WRT Invariant......Page 230
2.1.2 D2S1 Partition Function of T[M3] and WRT Invariant......Page 231
2.1.3 Superconformal Index of T[M3] and Its Factorization......Page 237
2.1.5 Topologically Twisted Index of T[M3]......Page 239
2.2.1 M3=S3......Page 240
2.3.1 Refined Superconformal Index......Page 244
2.3.2 Topologically Twisted Index of T[L(p,1)]......Page 246
3 Four-Manifolds and Topological Modular Forms......Page 247
3.1 From Six to Two Dimensions, via Four-Manifolds......Page 251
3.1.1 Flux Compactifications of 6d (1,0) Theories......Page 253
3.1.2 Anomaly Polynomial Reduction......Page 254
3.1.3 Turning on Flavor Symmetry Background......Page 255
3.1.4 A Toy Model......Page 258
6d (1,0) Hyper Multiplet......Page 260
6d (1,0) Vector Multiplet......Page 261
3.1.6 A Cure for Non-compactness: Equivariant Partition Functions......Page 262
3.1.7 Defect Group, Relativeness and Modularity Level......Page 263
3.1.8 Three-Manifolds......Page 267
3.2.1  N M5-Branes Probing C2/Zk Singularity......Page 268
(N,k)=(2,1) Theory on M4=12......Page 270
 (N,k)=(2,2) Theory on M4=12......Page 273
3.2.2 E-String Theory......Page 274
References......Page 275
Contents......Page 279
1 Introduction......Page 280
2 Elliptic Hypergeometric Integrals......Page 281
3 Properties of the Elliptic Gamma Function......Page 285
4 The Elliptic Beta Integral......Page 290
5 An Elliptic Extension of the Euler–Gauss Hypergeometric Function......Page 294
6 Multiple Elliptic Hypergeometric Integrals......Page 301
7 Rarefied Elliptic Hypergeometric Integrals......Page 304
8 An Integral Bailey Lemma......Page 308
9 Connection with Four Dimensional Superconformal Indices......Page 315
References......Page 323
Contents......Page 327
1 Introduction......Page 328
2 Feynman Integrals......Page 329
2.1 The Parametric Representation......Page 330
2.2 Maximal Cut......Page 332
2.3 The Differential Equations......Page 333
2.3.1 The Logarithmic Integral......Page 334
2.3.2  Elliptic Curve......Page 335
3 Toric Geometry and Feynman Graphs......Page 338
3.1 Toric Polynomials and Feynman Graphs......Page 339
3.1.1 Some Important Special Cases......Page 340
3.2 The GKZ Approach : A Review......Page 341
3.3.1 The Gauß Hypergeometric Series......Page 344
3.4.1 Maximal Cut......Page 345
3.4.2 The Differential Operator......Page 348
3.4.3 The Massive One-Loop Sunset Feynman Integral......Page 349
3.5 The Two-Loop Sunset......Page 350
3.5.1 The Differential Operators......Page 353
3.6.1 The All Equal Mass Case......Page 355
3.6.2 The General Mass Case......Page 356
3.7 Toward a Systematic Approach to Differential Equations......Page 360
4.1 The Sunset Integral as an Elliptic Dilogarithm......Page 361
4.2 The Sunset Integral as a Trilogarithm......Page 364
4.2.1 The All Equal Masses Case......Page 366
4.3.1 The Sunset Graph Polynomial and del Pezzo Surface......Page 367
4.3.2 Local Mirror Symmetry......Page 369
5 Conclusion......Page 370
References......Page 371
Contents......Page 376
2.1 The Differential Painlevé Equations......Page 377
2.2 Painlevé Test......Page 379
2.3 Isomonodromic Deformation......Page 380
2.4 Space of Initial Conditions......Page 381
2.5 QRT System......Page 382
2.6 ``Integrable'' Deautonomization......Page 385
2.7 Sakai Scheme......Page 386
2.8 Short Summary on Cubic Curve and Elliptic Function......Page 387
3.1 Affine Weyl Group and Its Translations......Page 390
3.2 Birational Representations......Page 392
3.3 Weyl Group Actions on Borel Subgroups......Page 394
3.4 Weyl Group Actions on Point Configurations......Page 396
3.5 The Elliptic Painlevé Equation in E10 Form......Page 398
3.6 Picard Lattice......Page 400
3.7 q-Painlevé Equation of Type E8(1)......Page 402
3.8 Toric-Like Form......Page 405
4.1 τ Functions......Page 408
4.2 Lax Formalism......Page 411
4.3 Hypergeometric Solution......Page 415
References......Page 419