Integrable Systems in Quantum Field Theory and Statistical Mechanics (eBook)

Front Cover 1
Integrable Systems in Quantum Field Theory and Statistical Mechanics 4
Copyright Page 5
Table of Contents 12
Foreword 8
Preface to the Present Volume 10
Chapter 1. Eigenvalue Spectrum of the Superintegrable Chiral Potts Model 14
1. Introduction 14
2. Formalism and the phase I ground state energy 21
3. Single particle excitation in phases I and IV 36
4. Multiparticle excitations for Q = 0 40
5. Discussion 50
Acknowledgement 53
Appendix A 54
Appendix B 57
Appendix C 59
References 67
Chapter 2. Onsager's Star-Triangle Equation: Master Key to Integrability 70
1. Introduction 70
2. Star-triangle equation 71
3. Chiral Potts model 79
4. Integrable quantum spin chain 91
Acknowledgements 96
Appendix 96
References 101
Chapter 3. Solving Models in Statistical Mechanics 108
Star-triangle relation 110
Difference property 112
Calculation of the free energy 112
Superintegrable chiral Potts model 123
References 127
Chapter 4. KdV-Type Equations and W-Algebras 130
Chapter 5. Boundary Conditions in Conformal Field Theory 140
1. Introduction 140
2. Boundary operators in conformal field theory 145
3. Conformal field theory on an annulus 148
4. Examples 153
5. Relation to the corner transfer matrix 155
6. Virasoro algebra in the non-critical Ising model 157
Acknowledgements 160
References 161
Chapter 6. Paths, Maya Diagrams and representations of s l ( r, C) 162
1. Introduction 162
2. Examples 165
3. The Fock representation of gl(8, C ) , gI(r, C) and sl(r, C) 173
4. Maya diagrams and Plücker relations 180
5. Paths and lifts 188
6. Paths and divisors 194
References 203
Chapter 7. Knot Theory based on Solvable Models at Criticality 206
1. Introduction 207
2. Yang-Baxter relation 210
3. Exactly solvable models 221
4. Knot theory 231
5. Exactly solvable models and knot theory 242
6. New link polynomials 258
7. Two-Variable Extension 274
8. Related Topics 284
9. Concluding Remarks 293
Acknowledgements 294
References 294
Chapter 8. From the Harmonic Oscillator to the A-D-E Classification of Conformal Models 300
1. Introduction 300
2. The harmonic oscillator 300
3. Free particle in a box 305
4. Rational billiards 311
5. The A-D-E classification of conformal models 320
6. Extensions 347
References 358
Chapter 9. Formal Groups and Conformal Field Theory over Z 360
0. Introduction 360
1. Formal groups 361
2. Construction of a formal group 365
3. Jacobian varieties and t-functions 366
4. Operators Fn and Vn 371
5. Zeta functions 375
References 378
Chapter 10. A New Family of Solvable Lattice Models Associated with An(1) 380
1. Introduction 380
2. Unrestricted models 390
3. Restricted models 393
4. Local state probabilities 394
Appendix A. Proof of Theorem 3 404
Appendix B. Proof of Theorem 4 406
Acknowledgements 409
Chapter 11. Solvable Lattice Models and Algebras of Face Operators 412
1. Introduction 412
2. Solvable lattice models 413
3. An algebra of face operators 418
4. An algebra of Yang-Baxter operators 420
5. Brauer's centralizer algebra and its q-analogue 422
6. Problems 426
References 427
Chapter 12. D-Modules and Nonlinear Systems 430
1. Tschirnhaus transformations for algebraic systems 430
2. Non-linear equations as non-commutative algebras 435
3 . Linearization 443
4. Microlocalization 444
Chapter 13. Quantum Groups and Integrable Models 448
1. History of the subject 448
2. Quantum matrix algebras 453
3. Quantum groups SLq(n) and GLq(n) 456
4. Quantum groups SOq(n) and Spq(n) 459
5. Quantum simple Lie algebras 463
6. Problems 467
References 468
Chapter 14. Conformal Field Theory on Universal Family of Stable Curves with Gauge Symmetries 472
Introduction 472
Notations 476
1. Integrable highest weight representation of affine Lie algebra 478
2. Pointed stable curves and the associated vacua 485
3. Universal family of pointed stable curves 511
4. Sheaf of vacua attached to local universal family 533
5. Integrable Connection with Regular Singularity 542
6. Locally freeness and factorization 556
Acknowledgments 576
References 576
Chapter 15. Yang-Baxter Algebras, Conformal Invariant Models and Quantum Groups 580
1. Yang-Baxter algebras 580
2. Physical realizations of Yang-Baxter algebras 592
3. The six vertex model and its descendants 600
4. Finite-size correctios from the Bethe-ansatz and conformal invariance 613
5. The light-cone lattice approach 627
6. Braid groups and quantum groups from Yang-Baxter algebras 644
References 649
Chapter 16. Integrable Field Theory from Conformal Field Theory 654
1. Introduction 654
2. Integrals of motion in CFT 658
3. Equations of motion in perturbed field theory 661
4. Degenerate fields and integrals of motion 667
5. Integrals of motion and scattering theory 673
6. Integrals of motion and S-matrix in critical Ising model with magnetic field 680
Acknowledgements 685
References 686
Chapter 17. Errata to Vertex Operators in Conformal Field Theory on P1 and Monodromy Representations of Braid Group in Advanced Studies in Pure Mathematics 16,1988 688
References 695