New Trends in Quantum Integrable Systems: Proceedings of the Infinite Analysis 09

CONTENTS......Page 16
PREFACE......Page 10
WORKSHOP PROGRAM......Page 12
LIST OF PARTICIPANTS......Page 14
1. Introduction......Page 19
2. Linear integral equations......Page 22
3. Conclusions......Page 27
References......Page 28
1. Introduction......Page 29
2.1. R-matrix and the monodromy matrix of type (1, 1⊗L)......Page 31
2.3. Monodromy matrices......Page 32
3.2. Temperley-Lieb algebra......Page 34
3.3. Basis vectors of spin-l=2 representation of Uq(sl2)......Page 35
3.5. Hermitian elementary matrices......Page 36
4.1. Higher-spin monodromy matrix of type (l, (2s)⊗Ns)......Page 37
4.2. Integrable spin-s Hamiltonians......Page 38
5.1. Spin-s local operators in terms of global operators......Page 39
5.3. Conjecture of the spin-s ground-state solution......Page 40
5.4. Correlation functions of the integrable spin-s XXZ model on a long finite chain
......Page 41
5.5. Multiple-integral representations of spin-s XXZ correlation function for arbitrary matrix elements......Page 43
6.1. Fundamental commutation relations......Page 45
6.2. Finite-sum expression of correlation functions for a finite chain......Page 47
Acknowledgments......Page 49
References......Page 50
1. Introduction......Page 53
1.1. Coinvariants......Page 54
1.2. Representation categories......Page 56
1.3. The main statement......Page 57
2. General facts about (1, p) models......Page 58
2.1. The doublet algebra......Page 59
2.2. Irreducible modules of A(p)......Page 60
3.2. Quasitensor structure......Page 63
4.1. Induced modules of A(p)......Page 65
4.2. Decompositions of induced A(p)-modules......Page 66
4.3. Characters of induced modules......Page 67
4.4. Abelianization......Page 68
5. Characters of multiplicity spaces......Page 70
5.1. Multiplicity spaces as coinvariants
......Page 72
5.2. Felder resolution......Page 74
6. Conclusions......Page 76
References......Page 77
1. Introduction......Page 79
2. Symmetric functions and Casoratians......Page 82
3. The XXZ spin-1/2 chain and the Algebraic Bethe Ansatz......Page 85
4. Continuous KP, Miwa variables and discrete KP......Page 89
5. Bethe scalar products are discrete KP τ-functions......Page 91
6. Remarks......Page 96
References......Page 97
1. Introduction......Page 99
2.1. The problem......Page 100
2.2. The Baxter-Luscher formula......Page 101
2.3. A summary of results for bulk quantities......Page 102
2.4. The 1D quantum partition function as a 2D classical partition function......Page 104
2.5. Commuting QTM......Page 106
3.1. Bethe roots......Page 108
3.2. Non-linear Integral Equation (NLIE)......Page 110
4. DME (density matrix elements) at finite temperatures......Page 113
References......Page 117
1. Introduction......Page 119
2.1. Good tropicalization of algebraic curves......Page 120
2.2. Smoothness of tropical curves......Page 121
2.4. Tropical analogue of Fay's trisecant identity......Page 122
2.5. Tropical Jacobian......Page 123
3.1. Generalized discrete periodic Toda lattice T(M,N)......Page 124
3.2. Generalized ultradiscrete periodic Toda lattice T(M,N)......Page 126
3.3. Spectral curves for T(M,N) and good tropicalization......Page 127
4.1. Bilinear equation......Page 128
4.2. Example: T(3, 2)......Page 129
4.3. Conjectures on T(M,N)......Page 131
References......Page 133
1. Introduction......Page 135
2. Two scaling limits of the XXZ model and two chiralities......Page 141
3. Inhomogeneous six vertex model and sine-Gordon model......Page 147
References......Page 153
PERFECT CRYSTALS FOR THE QUANTUM AFFINE ALGEBRA Uq(C(1)n)......Page 157
Introduction......Page 158
1. Quantum affine algebras and perfect crystals
......Page 159
2. The adjoint crystals of type C(1)n......Page 163
3. Kirillov-Reshetikhin crystals B1,2lA(2)2n+1 and B1,2lC(1)n......Page 167
4. Main Theorem......Page 171
References......Page 173
1. Introduction......Page 175
2. Boundary quantum Knizhnik-Zamolodchikov equation......Page 177
3.1. Affine Hecke algebra and Noumi representation
......Page 179
3.2. qKZ family......Page 181
3.3. Eigenvalue problem......Page 182
4.1. Non-symmetric Koornwinder polynomials......Page 184
4.3. Specialized case 1......Page 186
References......Page 188
1. Introduction......Page 191
1.1.2. Chari's graded g[t]-modules......Page 192
1.1.4. The combinatorial KR-conjecture: The M-sum formula......Page 193
1.1.6. Feigin-Loktev fusion products......Page 195
1.2. A pentagon of identities......Page 196
2.1. Finite-dimensional g[t]-modules and the fusion action......Page 197
2.2. Chari's KR-modules of g[t]......Page 198
3. Functional realization of fusion spaces......Page 199
3.1. Characterization of functions in Cλ,n......Page 200
3.2. Filtration of the space of functions......Page 202
4.1. The case of sl2......Page 204
4.2. The simply-laced case......Page 207
4.3. The non-simply laced case......Page 208
5. Summary......Page 209
References......Page 210
1. Introduction......Page 213
2.1. Integrability of the Toda chain......Page 215
2.2. Separation of variables......Page 216
3. Quantization conditions......Page 217
3.1. Gutzwiller's formulation of the quantization conditions......Page 218
3.2. Reformulation in terms of solutions to a nonlinear integral equation......Page 219
3.3. Solutions to the Baxter equation from the solutions to a NLIE......Page 221
3.4. Definition of Yang's potential
......Page 223
Appendix A.1. Analytic properties of Gutzwiller's solution......Page 224
Appendix A.2. Bounds for K+......Page 227
Appendix B. Existence and uniqueness of solutions to the non-linear integral equation......Page 229
Appendix C.1. Analytic properties of Q±δ
......Page 232
Appendix C.2. Baxter equation......Page 233
Appendix C.3. The quantization conditions......Page 234
References......Page 237
1. Introduction......Page 239
2.1. Crystals and combinatorial R......Page 241
2.2. Generalized local energies......Page 242
2.3. Generalized energies......Page 246
3.1. States and time evolution......Page 249
3.2. Counting particles and anti-particles......Page 251
4.1. Counting functions and generalized energies......Page 253
4.2. *-transformed correspondence......Page 255
5. Connection with combinatorial Bethe ansatz......Page 256
5.2. Conjecture on ultradiscrete tau functions......Page 257
References......Page 259
1. Introduction......Page 261
2. Painleve VI and JMU τ-function......Page 262
3. Hypergeometric kernel determinant......Page 264
4. PBT τ-function......Page 266
5.1. Basic notation......Page 268
5.2. Derivation......Page 269
6. Jimbo's asymptotic formula......Page 271
7. Asymptotics of D(t) as t → 1......Page 273
8. Numerics......Page 275
9. Special solutions check......Page 276
10.1. Flat space limit: PVI → PV......Page 279
10.2. Zero field limit: PV → PIII
......Page 280
Appendix A
......Page 282
References......Page 283
1. Introduction......Page 287
2. Definitions......Page 290
3.1. Shapes and combinatorics......Page 292
3.2. On representations of wreaths......Page 293
3.3. Useful decompositions of ΛΛ*......Page 295
3.4. Decomposing the regular Pxn-module......Page 296
3.5. Examples and combinatorial restriction......Page 301
3.6. Discussion......Page 302
References......Page 303
1. Introduction......Page 305
2. Form factor expansion and the λ extension......Page 307
3. Leading divergence as T → Tc......Page 309
5. The theta function expressions of Orrick, Nickel, Guttmann and Perk......Page 310
7. Diagonal form factors......Page 312
8. Nome q-representation versus modulus k-representation......Page 316
8.1. f(2n)0,0......Page 317
8.2. f(2n+1)0,0......Page 318
9. The λ generalized correlations
......Page 319
10. Diagonal susceptibility......Page 320
12. Conclusion......Page 322
References......Page 323
1. Introduction......Page 325
2.2. Master function......Page 327
2.3. Generators of the local algebra of a critical point......Page 328
2.4. Polynomials hi......Page 329
3.1. Algebra Oλ(∞)......Page 330
3.3. Isomorphism of algebras......Page 331
4.3. Definition of Bethe algebra
......Page 332
4.5. Shapovalov Form......Page 333
5.1. Definition of the weight function......Page 334
5.2. Grothendieck residue and Hessian......Page 335
5.4. Bethe ansatz......Page 336
5.5. Main result......Page 337
6.2. Bilinear form (,)S......Page 338
6.3. Auxiliary lemmas......Page 339
7. Concluding remarks......Page 340
References......Page 341
1. Introduction......Page 343
2. T and Y-systems......Page 344
3. Cluster algebra with coeffcients......Page 347
4.2. Parity decompositions of T and Y-systems......Page 348
4.3. Quiver Ql(Mt)......Page 349
4.4. Embedding maps......Page 352
4.5. T-system and cluster algebra......Page 355
4.6. Y-system and cluster algebra......Page 357
5.1. Parity decompositions of T and Y-systems......Page 358
5.2. Quiver Ql(Mt)......Page 359
5.3. Embedding maps......Page 361
6.1. Parity decompositions of T and Y-systems......Page 363
6.2. Construction of quiver Ql(C)......Page 365
6.3. Mutation sequence......Page 368
6.4. T-system, Y-system, and cluster algebra......Page 369
7.1. The case X'(C) is bipartite......Page 370
7.2. The case X'(C) is nonbipartite......Page 371
References......Page 372
1.1. Periodic Benjamin-Ono equation with discrete Laplacian......Page 375
1.2. Poisson algebra and Toda field equation......Page 376
2.1. Hirota-Miwa equation......Page 380
3.2. Integrals of motion from Ding-Iohara algebra......Page 383
3.3. Integrals of motion associated with t and t......Page 384
3.4. Formulas for M2 and M3......Page 385
3.5. Main conjecture and equations with respect to t2, t3......Page 386
References......Page 389
1. Introduction......Page 391
2.1. Schur functions......Page 392
2.2. Tau functions of KP hierarchy......Page 394
2.3. Tau functions of 2-KP hierarchy......Page 396
3.1. Setup of model......Page 397
3.2. Izergin-Korepin formula for ZN......Page 399
3.3. KP and 2-KP tau functions hidden in ZN......Page 400
4.1. L- and T-matrices for spin 1/2 chain......Page 402
4.2. Algebraic Bethe ansatz......Page 404
4.4. KP tau function hidden in Sn(u, v)......Page 405
5. Scalar product of states in models at q = 0......Page 406
References......Page 408
1. Introduction......Page 411
2. Convolution......Page 413
3.1. Middle convolution......Page 416
3.3. Index of rigidity......Page 418
3.4. Example......Page 419
4.1. Index of rigidity......Page 422
4.2. Subspace......Page 425
4.3. Middle convolution......Page 426
4.4. Classification......Page 430
5. Concluding remarks......Page 436
References......Page 438
1. Introduction......Page 439
2. Boundary rational qKZ equation......Page 441
3.1. Commuting dierential operators......Page 442
3.2. Compatibility......Page 444
4.1. The double affine Hecke algebra of type (Cvn, Cn)
......Page 448
4.2. The bispectral qKZ equation......Page 450
4.3. The degenerate double affine Hecke algebra......Page 453
4.4. Degeneration of the bispectral qKZ equation (Y-side)
......Page 455
4.5. Degeneration of the bispectral qKZ equation (X-side)......Page 456
4.6. Embedding into the compatible system......Page 457
5. Integral formula for solutions in a special case......Page 461
References......Page 467
1. Introduction......Page 469
2. Critical ZN-symmetric vertex model......Page 471
3. Reection equation......Page 472
4. Summary of results in the previous paper and notations......Page 473
5. Consequences of Lemma 4.4......Page 477
6. Type I components of the reection equation......Page 480
6.1. Relations between diagonal and non-diagonal elements of K-matrix......Page 482
7. Determination of the diagonal elements......Page 485
8. Determination of the off diagonal elements......Page 498
9. Relations among parameters......Page 504
10. Main results......Page 510
References......Page 516