New trends in quantum integrable systems : proceedings of the Infinite Analysis 09 : Kyoto, Japan 27-31 July 2009

The present volume is the result of the international workshop on New Trends in Quantum Integrable Systems that was held in Kyoto, Japan, from 27 to 31 July 2009. As a continuation of the RIMS Research Project "Method of Algebraic Analysis in Integrable Systems" in 2004, the workshop's aim was to cover exciting new developments that have emerged during the recent years. Collected here are research articles based on the talks presented at the workshop, including the latest results obtained thereafter. The subjects discussed range across diverse areas such as correlation functions of solvable models, integrable models in quantum field theory, conformal field theory, mathematical aspects of Bethe ansatz, special functions and integrable differential/difference equations, representation theory of infinite dimensional algebras, integrable models and combinatorics. Through these topics, the reader can learn about the most recent developments in the field of quantum integrable systems and related areas of mathematical physics. 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 di erential 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 Properties of linear integral equations related to the six-vertex model with disorder parameter / Hermann Boos and Frank Gohmann -- Algebraic aspects of the correlation functions of the integrable higher-spin XXZ spin chains with arbitrary entries / Tetsuo Deguchi and Chihiro Matsui -- Characters of coinvariants in (1,p) logarithmic models / Boris L. Feigin and Ilya Yu. Tipunin -- XXZ scalar products, Miwa variables and discrete KP / Omar Foda and Gus Schrader -- Quantum spin chains at finite temperatures / Frank Gohmann and Junji Suzuki -- Tropical spectral curves, Fay's trisecant identity, and generalized ultradiscrete Toda lattice / Rei Inoue and Shinsuke Iwao -- On one-point functions of descendants in sine-Gordon model / Michio Jimbo, Tetsuji Miwa and Fedor Smirnov -- Perfect crystals for the quantum affine algebra U[symbol] / Seok-Jin Kang [und weitere] -- Boundary quantum Knizhnik-Zamolodchikov equation / Masahiro Kasatani -- A pentagon of identities, graded tensor products, and the Kirillov-Reshetikhin conjecture / Rinat Kedem -- TBA for the Toda chain / Karol Kajetan Kozlowski and Jorg Teschner -- Generalized energies and integrable D[symbol] cellular automaton / Atsuo Kuniba, Reiho Sakamoto and Yasuhiko Yamada -- Dyson's constant for the hypergeometric kernel / Oleg Lisovyy -- Representation theory of a small ramified partition algebra / Paul P. Martin -- The saga of the Ising susceptibility / Barry M. McCoy -- Bethe algebra of the [symbol] Gaudin model and algebra of functions on the critical set of the master function / Evgeny Mukhin, Vitaly Tarasov and Alexander Varchenko -- T-systems, Y-systems, and cluster algebras : Tamely laced case / Tomoki Nakanishi -- Periodic Benjamin-Ono equation with discrete Laplacian and 2D-Toda hierarchy / Jun'ichi Shiraishi and Yohei Tutiya -- KP and Toda tau functions in Bethe ansatz / Kanehisa Takasaki -- Introduction to middle convolution for differential equations with irregular singularities / Kouichi Takemura -- Differential equations compatible with boundary rational qKZ equation / Yoshihiro Takeyama -- Classification of solutions to the reflection equation for the critical Z[symbol]-symmetric vertex model I / Yuji Yamada The present volume is the result of the international workshop on New Trends in Quantum Integrable Systems that was held in Kyoto, Japan, from 27 to 31 July 2009. As a continuation of the RIMS Research Project Method of Algebraic Analysis in Integrable Systems in 2004, the workshop's aim was to cover exciting new developments that have emerged during the recent years.Collected here are research articles based on the talks presented at the workshop, including the latest results obtained thereafter. The subjects discussed range across diverse areas such as correlation functions of solvable models, integrable models in quantum field theory, conformal field theory, mathematical aspects of Bethe ansatz, special functions and integrable differential/difference equations, representation theory of infinite dimensional algebras, integrable models and combinatorics.Through these topics, the reader is exposed to the most recent developments in the field of quantum integrable systems and related areas of mathematical physics. A volume that is the result of the international workshop on New Trends in Quantum Integrable Systems that was held in Kyoto, Japan, from 27 to 31 July 2009. It includes research articles based on the talks presented at the workshop, including the results obtained thereafter.