Quantum groups in two-dimensional physics:
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge
Cambridge Univ. Press
1996
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | Cambridge monographs on mathematical physics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVIII, 457 S. graph. Darst. |
| ISBN: | 0521460654 |
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Datensatz im Suchindex
| _version_ | 1804125210294616064 |
|---|---|
| adam_text | Contents
Preface xv
1 S matrices, spin chains and vertex models 1
1.1 Factorized S matrix models 1
1.1.1 Zamolodchikov algebra 5
1.1.2 Example 7
1.2 Bethe s diagonalization of spin chain hamiltonians 10
1.3 Integrable vertex models: the six vertex model 14
Exercises 25
Appendix A Form factors 28
Al.l Introduction to Smirnov s program 28
A1.2 Form factors at work: the Ising model 31
Exercise 33
2 The Yang Baxter equation: a first look 34
2.1 The Yang Baxter algebra 34
2.1.1 The ^ matrix and the Yang Baxter equation 34
2.1.2 The monodromy matrix 38
2.1.3 Co product and the Yang Baxter algebra 39
2.1.4 Algebraic Bethe ansatz 41
2.2 Yang Baxter algebras and braid groups 47
2.3 Yang Baxter algebras and quantum groups 51
2.3.1 The ^ matrix as an intertwiner 53
2.3.2 A first contact with affine Hopf algebras 55
2.4 Descendants of the six vertex model 58
2.4.1 Descent procedure 58
2.4.2 Bethe ansatz for descendant models 61
2.5 Comments 66
2.5.1 Explanation of our conventions 66
2.5.2 On parametrizations of the six vertex weights 67
Exercises 67
3 Bethe ansatz: some examples 71
3.1 Introduction and summary 71
3.2 The phase structure of the six vertex model 73
ix
x Contents
3.3 Low lying excitations 78
3.3.1 Strings and holes 78
3.3.2 Dispersion relations 83
3.4 Integrable higher spin chains 83
3.4.1 Hidden symmetry 85
3.5 Integrability in a box: open boundary conditions 88
3.5.1 Vertex models: the Sklyanin equation 88
3.5.2 Spin chains: the Bethe ansatz 92
3.6 Hamiltonians with quantum group invariance 96
3.7 Spin 1 chains 98
Exercises 102
4 The eight vertex model 108
4.1 Definitions and Yang Baxter relations 108
4.2 Bethe ansatz solution 111
4.2.1 A smart change of basis 112
4.2.2 Eight vertex Yang Baxter algebra 116
4.3 Reference state and 0 parameter 118
4.3.1 Further comments on the 6 parameter 120
Exercises 121
Appendix B Elliptic functions 125
Exercises 128
Appendix C Sklyanin algebra 130
Exercises 132
5 Face models 134
5.1 Weights and graphs: the definitions 134
5.2 Trigonometric solutions 137
5.2.1 Bratelli diagrams 137
5.2.2 Yang Baxter operators 138
5.2.3 The Temperley Lieb Jones algebra 139
5.2.4 Towers of algebras associated with a graph 140
5.2.5 The algebra of face observables 144
5.2.6 The trigonometric solution for Coxeter models 146
5.3 Elliptic solutions 148
5.3.1 An example: the Ising model 148
5.3.2 Restricted and unrestricted models 154
5.4 Fusion for face models 155
5.5 The corner transfer matrix 158
Exercises 160
Appendix D Knots and integrable models 165
D5.1 Introduction: the Jones polynomial 165
D5.2 Markov moves 168
D5.3 Markov traces for the Hecke algebra 170
Contents xi
D5.4 The Burau representation 172
D5.5 Extended Yang Baxter systems 173
Exercises 176
Appendix E Spin models 177
E5.1 Factors and subfactors 177
E5.2 Spin models 180
Exercises 183
6 Quantum groups: mathematical review 184
6.1 Hopf algebras 184
6.2 Quasi triangular Hopf algebras 186
6.3 Drinfeld s quantum double 187
6.4 The quantum group Uq(stf(2)) 189
6.4.1 Quantum double construction 189
6.4.2 Irreducible representations 194
6.5 Centralizer and Hecke algebra 201
6.5.1 Representations of Hn(q) 203
6.6 Link invariants from quantum groups 205
6.7 The quantum group Uq(^0) 207
6.8 R matrices: an incomplete catalog 208
6.9 Classical Yang Baxter equation 210
6.10 Affine quantum groups 211
6.11 Quasi Hopf algebras 218
Exercises 219
7 Integrable models at roots of unity 228
7.1 Mathematical preliminaries 228
7.1.1 The center of Uq(st(2)) 228
7.1.2 Finite dimensional irreps 229
7.1.3 The co adjoint action 231
7.1.4 Intertwiners 234
7.2 A family of R matrices 235
7.2.1 Highest weight intertwiner 235
7.2.2 The nilpotent K matrix 238
7.3 Nilpotent hamiltonians 240
7.4 Bethe ansatz 244
7.5 The limit i oo 250
7.5.1 Quantum harmonic oscillators 251
7.5.2 Link invariants 252
7.6 The chiral Potts model 252
7.6.1 Star triangle relations 255
7.6.2 The associated spin chain hamiltonian 258
7.6.3 Self dual chiral Potts models 260
7.6.4 Super integrable chiral Potts models 262
xii Contents
7.6.5 The quantum symmetry 263
7.7 Solving the Yang Baxter equation 268
Exercises 269
8 Two dimensional conformal field theories 272
8.1 Introduction: critical phenomena 272
8.2 Renormalization group 272
8.3 Examples 275
8.3.1 The one dimensional Ising model 275
8.3.2 The gaussian model 278
8.4 Operator algebra of a universality class 280
8.5 Conformal invariance and statistical mechanics 281
8.6 The two dimensional conformal group 282
8.7 Representations of the Virasoro algebra 286
8.8 Decoupling of null vectors 291
8.8.1 The Kac formula 292
8.8.2 Conformal Ward identities 294
8.8.3 Minimal models 295
8.9 Fusion algebra 299
8.10 Finite size effects 300
Exercises 304
9 Duality in conformal field theories 308
9.1 Monodromy invariance 309
9.2 Conformal blocks and chiral vertex operators 311
9.3 Sewing 314
9.4 Braiding and fusion 319
9.5 Conformal field theories and towers of algebras 323
9.6 Genus one polynomial equations 327
Exercises 336
10 Coulomb gas representation 340
10.1 Free and Feigin Fuks scalar fields 340
10.2 Screening charges in correlation functions 344
10.2.1 Braiding matrices: an explicit example 348
10.2.2 Contour techniques 350
10.3 Lagrangian approach 353
10.4 Wess Zumino models 355
10.4.1 The Knizhnik Zamolodchikov equation 355
10.4.2 Free field representation of Wess Zumino models 359
10.4.3 The Goddard Kent Olive construction 363
10.5 Magic corner transfer matrix 365
Exercises 366
Appendix F Vertex operators 373
Contents xiii
11 Quantum groups in conformal field theory 376
11.1 The hidden quantum symmetry 376
11.2 Braiding matrices and quantum 6j symbols 381
11.3 Ribbon Hopf algebras 384
11.4 The contour representation of quantum groups 386
11.4.1 Screened vertex operators 386
11.4.2 Examples 390
11.4.3 The quantum qroup 392
11.4.4 The ^ matrix 396
11.4.5 Chiral vertex operators 400
11.5 The quantum group of SlJ(2)k 401
11.5.1 The ^ matrix 407
11.5.2 Fusion rules and chiral vertex operators 409
11.5.3 On intertwiners: a clarification 413
11.6 The quantum group of minimal models 413
Exercises 415
Appendix G Super conformal field theories 422
Gll.l Super conformal transformations 422
G11.2 Representations 424
GU.3 N = 2 super conformal algebras 425
G11.4 N = 2 irreps and the chiral ring 426
Gl 1.5 N = 2 topological theories 429
Gil.6 Perturbed chiral ring 430
G11.7 Landau Ginsburg description 432
G11.8 Quantum groups and solitons 434
Exercise 437
References 439
Index 452
|
| any_adam_object | 1 |
| author | Gómez, César Ruiz-Altaba, Marti Sierra, Germán 1955- |
| author_GND | (DE-588)115395547 |
| author_facet | Gómez, César Ruiz-Altaba, Marti Sierra, Germán 1955- |
| author_role | aut aut aut |
| author_sort | Gómez, César |
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| building | Verbundindex |
| bvnumber | BV010732372 |
| callnumber-first | Q - Science |
| callnumber-label | QC20 |
| callnumber-raw | QC20.7.G76 |
| callnumber-search | QC20.7.G76 |
| callnumber-sort | QC 220.7 G76 |
| callnumber-subject | QC - Physics |
| classification_rvk | UK 3000 |
| classification_tum | PHY 012f |
| ctrlnum | (OCoLC)32820491 (DE-599)BVBBV010732372 |
| dewey-full | 530.1/43/0151255 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.1/43/0151255 |
| dewey-search | 530.1/43/0151255 |
| dewey-sort | 3530.1 243 6151255 |
| dewey-tens | 530 - Physics |
| discipline | Physik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV010732372 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T17:57:56Z |
| institution | BVB |
| isbn | 0521460654 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007165453 |
| oclc_num | 32820491 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-91G DE-BY-TUM DE-384 DE-703 DE-19 DE-BY-UBM DE-634 DE-188 |
| owner_facet | DE-355 DE-BY-UBR DE-91G DE-BY-TUM DE-384 DE-703 DE-19 DE-BY-UBM DE-634 DE-188 |
| physical | XVIII, 457 S. graph. Darst. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series2 | Cambridge monographs on mathematical physics |
| spelling | Gómez, César Verfasser aut Quantum groups in two-dimensional physics César Gómez ; Martí Ruiz-Altaba ; Germán Sierra 1. publ. Cambridge Cambridge Univ. Press 1996 XVIII, 457 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge monographs on mathematical physics Conforme invarianten gtt Groupes quantiques ram Invariants conformes ram Kwantumgroepen gtt Physique mathématique ram Yang-Baxter, Équation de ram Yang-Baxter-vergelijkingen gtt Mathematische Physik Conformal invariants Mathematical physics Quantum groups Yang-Baxter equation Quantengruppe (DE-588)4252437-4 gnd rswk-swf Dimension 2 (DE-588)4321721-7 gnd rswk-swf Vertexalgebra (DE-588)4328736-0 gnd rswk-swf Quantenmechanik (DE-588)4047989-4 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Konforme Differentialgeometrie (DE-588)4206468-5 gnd rswk-swf Zweidimensionale Raum-Zeit (DE-588)4303349-0 gnd rswk-swf Yang-Baxter-Gleichung (DE-588)4291478-4 gnd rswk-swf Konforme Feldtheorie (DE-588)4312574-8 gnd rswk-swf Bethe-Ansatz (DE-588)4121011-6 gnd rswk-swf Vertexoperator (DE-588)4188067-5 gnd rswk-swf Integrierbarkeit (DE-588)4474751-2 gnd rswk-swf Deformation Mathematik (DE-588)4011284-6 gnd rswk-swf Yang-Baxter-Gleichung (DE-588)4291478-4 s Bethe-Ansatz (DE-588)4121011-6 s Mathematische Physik (DE-588)4037952-8 s DE-604 Quantengruppe (DE-588)4252437-4 s Konforme Feldtheorie (DE-588)4312574-8 s Zweidimensionale Raum-Zeit (DE-588)4303349-0 s Vertexoperator (DE-588)4188067-5 s Vertexalgebra (DE-588)4328736-0 s Konforme Differentialgeometrie (DE-588)4206468-5 s Dimension 2 (DE-588)4321721-7 s 1\p DE-604 Quantenmechanik (DE-588)4047989-4 s Deformation Mathematik (DE-588)4011284-6 s Integrierbarkeit (DE-588)4474751-2 s 2\p DE-604 3\p DE-604 4\p DE-604 Ruiz-Altaba, Marti Verfasser aut Sierra, Germán 1955- Verfasser (DE-588)115395547 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007165453&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 4\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Gómez, César Ruiz-Altaba, Marti Sierra, Germán 1955- Quantum groups in two-dimensional physics Conforme invarianten gtt Groupes quantiques ram Invariants conformes ram Kwantumgroepen gtt Physique mathématique ram Yang-Baxter, Équation de ram Yang-Baxter-vergelijkingen gtt Mathematische Physik Conformal invariants Mathematical physics Quantum groups Yang-Baxter equation Quantengruppe (DE-588)4252437-4 gnd Dimension 2 (DE-588)4321721-7 gnd Vertexalgebra (DE-588)4328736-0 gnd Quantenmechanik (DE-588)4047989-4 gnd Mathematische Physik (DE-588)4037952-8 gnd Konforme Differentialgeometrie (DE-588)4206468-5 gnd Zweidimensionale Raum-Zeit (DE-588)4303349-0 gnd Yang-Baxter-Gleichung (DE-588)4291478-4 gnd Konforme Feldtheorie (DE-588)4312574-8 gnd Bethe-Ansatz (DE-588)4121011-6 gnd Vertexoperator (DE-588)4188067-5 gnd Integrierbarkeit (DE-588)4474751-2 gnd Deformation Mathematik (DE-588)4011284-6 gnd |
| subject_GND | (DE-588)4252437-4 (DE-588)4321721-7 (DE-588)4328736-0 (DE-588)4047989-4 (DE-588)4037952-8 (DE-588)4206468-5 (DE-588)4303349-0 (DE-588)4291478-4 (DE-588)4312574-8 (DE-588)4121011-6 (DE-588)4188067-5 (DE-588)4474751-2 (DE-588)4011284-6 |
| title | Quantum groups in two-dimensional physics |
| title_auth | Quantum groups in two-dimensional physics |
| title_exact_search | Quantum groups in two-dimensional physics |
| title_full | Quantum groups in two-dimensional physics César Gómez ; Martí Ruiz-Altaba ; Germán Sierra |
| title_fullStr | Quantum groups in two-dimensional physics César Gómez ; Martí Ruiz-Altaba ; Germán Sierra |
| title_full_unstemmed | Quantum groups in two-dimensional physics César Gómez ; Martí Ruiz-Altaba ; Germán Sierra |
| title_short | Quantum groups in two-dimensional physics |
| title_sort | quantum groups in two dimensional physics |
| topic | Conforme invarianten gtt Groupes quantiques ram Invariants conformes ram Kwantumgroepen gtt Physique mathématique ram Yang-Baxter, Équation de ram Yang-Baxter-vergelijkingen gtt Mathematische Physik Conformal invariants Mathematical physics Quantum groups Yang-Baxter equation Quantengruppe (DE-588)4252437-4 gnd Dimension 2 (DE-588)4321721-7 gnd Vertexalgebra (DE-588)4328736-0 gnd Quantenmechanik (DE-588)4047989-4 gnd Mathematische Physik (DE-588)4037952-8 gnd Konforme Differentialgeometrie (DE-588)4206468-5 gnd Zweidimensionale Raum-Zeit (DE-588)4303349-0 gnd Yang-Baxter-Gleichung (DE-588)4291478-4 gnd Konforme Feldtheorie (DE-588)4312574-8 gnd Bethe-Ansatz (DE-588)4121011-6 gnd Vertexoperator (DE-588)4188067-5 gnd Integrierbarkeit (DE-588)4474751-2 gnd Deformation Mathematik (DE-588)4011284-6 gnd |
| topic_facet | Conforme invarianten Groupes quantiques Invariants conformes Kwantumgroepen Physique mathématique Yang-Baxter, Équation de Yang-Baxter-vergelijkingen Mathematische Physik Conformal invariants Mathematical physics Quantum groups Yang-Baxter equation Quantengruppe Dimension 2 Vertexalgebra Quantenmechanik Konforme Differentialgeometrie Zweidimensionale Raum-Zeit Yang-Baxter-Gleichung Konforme Feldtheorie Bethe-Ansatz Vertexoperator Integrierbarkeit Deformation Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007165453&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT gomezcesar quantumgroupsintwodimensionalphysics AT ruizaltabamarti quantumgroupsintwodimensionalphysics AT sierragerman quantumgroupsintwodimensionalphysics |