Integrability: from statistical systems to Gauge theory: lecture notes of the Les Houches Summer School: volume 106, 6 June-1 July 2016

Previous sessions......Page 6
Contents......Page 14
 Lecturers......Page 16
 Students and Auditors......Page 18
1 Integrability in statistical physics and quantum spin chains......Page 24
 1.1 Potts model......Page 26
 1.2 Quantum integrability......Page 37
 1.3 Algebraic Bethe ansatz......Page 47
 1.4 Thermodynamic limit......Page 59
 1.5 Nineteen-vertex models......Page 73
 References......Page 81
2 A guide to two-dimensional conformal field theory......Page 83
 Preface......Page 85
 2.1 Conformal symmetry......Page 86
 2.2 Bootstrap......Page 103
 2.3 Relations to integrable models......Page 124
 References......Page 140
3 Lectures on the holographic duality of gauge fields and strings......Page 144
 3.1 Preamble about gauge fields......Page 146
 3.2 The Wilson loop......Page 152
 3.3 The large N expansion......Page 157
 3.4 D branes, black branes, and the Maldacena conjecture......Page 165
 3.5 N = 4 superconformal quantum field theory......Page 173
 3.6 Holographic Wilson loops......Page 183
 References......Page 202
4 Introduction to scattering amplitudes......Page 206
 Preface......Page 208
 4.1 Kinematics......Page 209
 4.2 Amplitudes......Page 213
 4.3 Factorization......Page 217
 4.4 On-shell (BCFW) recursion relations......Page 219
 References......Page 227
5 Integrability in sigma-models......Page 228
 5.1 Introduction......Page 230
 5.2 Geometry......Page 231
 5.3 Principal chiral field......Page 235
 5.4 Symmetric cosets......Page 239
 5.5 B-field and topology......Page 244
 5.6 Quantum sigma-models......Page 247
 5.7 O(N) model: large N......Page 251
 5.8 O(N) model: exact solution......Page 253
 5.9 Crash course in superalgebras......Page 261
 5.10 Supercoset sigma-models......Page 264
 Acknowledgements......Page 268
 References......Page 269
6 Integrability in 2D fields theory/sigma-models......Page 271
 6.1 Liouville theorem......Page 273
 6.2 Free particle on a group manifold......Page 276
 6.3 Non-linear sigma model......Page 278
 6.4 Principal chiral field......Page 281
 6.5 Integrable examples of NLSM......Page 283
 6.6 The cigar......Page 288
 6.7 B-field......Page 298
 6.8 Integrable examples of NLSM with B-field......Page 301
 6.9 Wick rotation......Page 302
 6.10 Free scalar field on a cylinder......Page 304
 6.11 General structure of the Hilbert space of 2D CFT......Page 308
 6.12 Dimensional transmutation in PCF......Page 310
 6.13 RG flow equations......Page 313
 6.14 String equations......Page 316
 6.15 The quantum cigar......Page 317
 6.16 The quantum sausage......Page 331
 6.17 Other integrable structures in the cigar NLSM......Page 337
 References......Page 340
7 Applications of integrable models in condensed matter and cold atom physics......Page 342
 7.1 Physical realizations of integrable models......Page 344
 7.2 Eigenstates of integrable and generic local Hamiltonians......Page 354
 7.3 Macro states in the spin-1/2 Heisenberg chain......Page 357
 7.4 Quantum quenches......Page 369
 References......Page 372
8 Introduction to integrability and one-point functions in N = 4 supersymmetric Yang–Mills theory and its defect cousin......Page 375
 8.1 Introduction......Page 377
 8.2 N = 4 supersymmetric Yang–Mills theory and thespectral problem......Page 379
 8.3 The integrable Heisenberg spin chain in N = 4supersymmetric Yang–Mills theory......Page 395
 8.4 N = 4 supersymmetric Yang–Mills theory with a defect,one-point functions, and integrability......Page 403
 8.5 Outlook......Page 417
 References......Page 418
9 Spectrum of N = 4 supersymmetric Yang–Mills theory and the quantum spectral curve......Page 423
 9.2 From harmonic oscillator to QQ-relations......Page 425
 9.3 Classical string and strong coupling limit of QSC......Page 441
 9.4 QSC formulation......Page 447
 9.5 QSC—analytic examples......Page 453
 9.6 Solving QSC at finite coupling numerically......Page 460
 9.7 Applications, further reading, and open questions......Page 466
 References......Page 468
10 Three-point functions in N = 4 supersymmetric Yang–Mills theory......Page 472
 10.1 Three-point functions at weak coupling......Page 474
 10.2 Hexagon approach......Page 495
 10.3 Conclusion and prospects......Page 521
 References......Page 522
11 Localization and N = 2 supersymmetric field theory......Page 524
 11.1 Gauge theory......Page 526
 11.2 Kähler, special Kähler, and hyperKähler geometry......Page 530
 11.3 Supersymmetry......Page 535
 11.4 Supersymmetric gauge theory......Page 540
 11.5 Seiberg–Witten integrable system......Page 554
 References......Page 562
Appendix A Lectures given at the school but no written contribution......Page 564
Appendix B Seminars given during the school......Page 570