Non-Perturbative Methods in 2 Dimensional Quantum Field Theory (Second Edition)

Preface\\r......Page 6
Contents......Page 10
1 Introduction......Page 18
2.2 Bosonic Free Fields......Page 26
2.3 Fermionic Free Fields......Page 30
2.4 Bosonization of Massless Fermions......Page 32
2.5 The RS-Model......Page 36
2.6 Conclusions......Page 39
3.1 Introduction......Page 42
3.2 The Massless Thirring Model......Page 43
3.3.1 Equivalence with sine-Gordon equation......Page 46
3.3.2 Classical conservation laws......Page 48
3.3.3 Quantum conservation laws......Page 49
3.4.1 Fermions in terms of bosons......Page 54
3.5 The Soliton as a Disorder Parameter......Page 56
3.6 Conclusion......Page 62
4.1 Introduction......Page 66
4.2 Functional Determinant one-loop diagram......Page 67
4.2.1 Determinants and the Generalized Zeta-Function......Page 70
4.2.2 One Point Compactification......Page 76
4.2.3 The associated Dirac operator......Page 79
4.3.1 The perturbative approach......Page 82
4.3.2 The Schwinger-DeWitt method......Page 84
4.3.3 The Fujikawa method......Page 87
4.4.1 C-function regularization......Page 89
4.4.2 Proper-time regularization......Page 91
4.4.3 The Fujikawa point of view......Page 92
4.5 A Theorem on a one parameter family of factorizable operators......Page 96
4.6 The QCD2 functional determinant......Page 99
4.7.1 Axial anomaly equation in the presence of zero-modes......Page 102
4.7.2 Atiyah-Singer Index Theorem......Page 105
4.8.1 Ambiguities in the regularization......Page 108
4.8.2 Dependence on the scale parameter......Page 109
4.9 Mass expansion in proper-time regularization......Page 112
4.10 The Finite Temperature Heat Kernel......Page 116
4.10.1 Scalar held in a static background potential......Page 118
4.10.2 Scalar field in a static background gauge potential......Page 120
4.11 Conclusion......Page 124
5.2 The O(N) Invariant Gross-Neveu Model......Page 128
5.2.1 Classical conservation laws......Page 129
5.2.2 Effective potential and B-function in a 1/N expansion......Page 130
5.2.3 The 1/N Expansion: Feynman rules......Page 134
5.2.4 Leading order S-matrix elements......Page 136
5.2.5 Quantization of the non-local charge......Page 139
5.3 Chiral Gross-Neveu Model......Page 142
5.3.1 Cancellation of infrared singularities......Page 143
5.3.2 The 1/N expansion......Page 145
5.3.3 Operator formulation......Page 147
5.3.4 Quantization of non-local charge......Page 152
5.4 Conclusions and Physical Interpretation......Page 153
6.1 Historical development......Page 156
6.2 Sigma models and current algebra......Page 157
6.3 Two-dimensional o models: preliminaries......Page 159
6.4.1 Formal developments......Page 167
6.4.2 Dual symmetry and higher conservation laws......Page 173
6.4.3 An explicit example: the Grassmannians......Page 183
6.5.1 Definition and properties......Page 185
6.5.2 Dual symmetry and higher conservation laws......Page 189
6.5.3 Construction of an explicit example......Page 196
6.6 Analogies with 4D Gauge Theories......Page 201
6.7 Concluding Remarks......Page 205
7.1 Introduction......Page 212
7.2.1 1/N expansion......Page 213
7.2.2 Renormalization......Page 219
7.2.3 Infrared divergencies......Page 220
7.2.4 Physical interpretation of the results......Page 222
7.3.1 1/N expansion and Feynman rules......Page 223
7.3.2 Physical interpretation of the results......Page 228
7.4.1 Purely bosonic sigma models and anomalies......Page 234
7.4.2 Fermionic interaction and anomaly cancellation......Page 240
7.5.1 Bosonic O(N)-symmetric sigma models......Page 243
7.6 Non-local charges in the WZNW model......Page 255
7.7.1 Background Field Method......Page 258
7.7.2 Parallelizable manifolds; applications to string theory......Page 264
7.8 Anomalous Non-Linear o Models in four dimensions......Page 266
7.9 Conclusion......Page 267
8.1.1 Consequences of higher conservation laws......Page 274
8.1.2 Factorizable S-matrix......Page 275
8.1.3 Fusion rules......Page 279
8.2.1 SU(N) invariant S-matrices......Page 281
8.2.2 Sine-Gordon and massive Thirring models......Page 283
8.2.3 Exact S-matrix for O(N) symmetry......Page 288
8.2.4 The ZN invariant S-matrix......Page 289
8.3.1 S-matrices of purely fermionic models......Page 290
8.3.2 S-matrices of non-linear sigma models......Page 294
8.4 Boundary S-matrices......Page 304
8.5 Further Developments......Page 308
8.6 Conclusion......Page 309
9.1 Introduction......Page 314
9.2 Existence of a Critical Point......Page 316
9.3 Properties at the Critical Point......Page 319
9.3.1 The Polyakov-Wiegmann formula......Page 320
9.3.2 The Affine algebra......Page 321
9.3.3 The WZW fields in terms of fermions......Page 323
9.3.4 The Sugawara form of the energy-momentum tensor......Page 324
9.3.5 The non-Abelian bosonization in the operator language......Page 325
9.4 Properties off the Critical Point......Page 326
9.4.1 Integrability of the WZNW action......Page 327
9.4.2 On the solution off the critical point......Page 328
9.4.3 Supersymmetric WZW model......Page 330
9.5 Conclusion......Page 332
10.1 Introduction......Page 334
10.2.1 Quantum solution......Page 336
10.2.2 The Maxwell current......Page 338
10.2.3 Chiral densities......Page 341
10.2.4 Vacuum structure......Page 342
10.2.5 Gauge transformations......Page 346
10.2.6 Correlation functions and violation of clustering......Page 349
10.2.7 Absence of charged states (screening)......Page 350
10.2.8 The quark-antiquark potential......Page 352
10.2.9 Adding flavour......Page 354
10.2.10 Fractional winding number and the U(1) problem......Page 357
10.3.1 Equivalent bosonic formulation......Page 361
10.3.2 The quantum Dirac equation......Page 363
10.3.3 Vacuum structure and all that......Page 366
10.3.4 Screening versus confinement......Page 367
10.3.5 Adding flavour......Page 375
10.3.6 Lorentz transformation properties......Page 381
10.3.7 The MSM as the limit of a massive vector theory......Page 384
10.4 Conclusion......Page 387
11.1 Introduction......Page 392
11.2 The 1/N expansion: \'t Hooft model......Page 395
11.3 Currents Green functions and determinants......Page 400
11.3.1 Tree graph expansion of the current......Page 401
11.3.2 Recovering the QCD2 effective action......Page 403
11.3.3 Fermion Green Function......Page 406
11.4 Local decoupled formulation and BRST constraints......Page 409
11.4.1 Local decoupled partition function and BRST symmetries......Page 410
11.4.2 Systematic derivation of the constraints......Page 415
11.5.1 Non-local decoupled partition function and BRST symmetries......Page 418
11.6 The physical Hilbert space......Page 422
11.7 The QCD2 vacuum......Page 423
11.8 Massive two-dimensional QCD......Page 426
11.9 Screening in two-dimensional QCD......Page 428
11.10 Further algebraic aspects......Page 434
11.11 Conclusions......Page 435
12.1 Introduction......Page 440
12.2 Equivalent Bosonic Action......Page 441
12.3.1 The external field current and chiral densities......Page 442
12.4 Vacuum Structure......Page 443
12.4.1 Chirality of the vacuum......Page 444
12.5 Why Study Gauge-Invariant Correlators......Page 448
12.6 Screening versus Confinement......Page 449
12.7 Quasi-Periodic Boundary Conditions and the 0-Vacuum......Page 451
12.8 Axial anomaly and the Dirac sea......Page 455
12.9 Functional Representation of Tunneling Amplitudes......Page 457
12.10 Interpretation of the Result......Page 459
12.10.1 Zero modes......Page 461
12.10.2 Calculation of det i D from the anomaly equation......Page 463
12.11 Eigenvalue Spectrum of the Dirac Operator......Page 465
12.12 Zero Modes and Boundary-Value Problem......Page 468
12.12.1 Free Dirac operator and non-local boundary conditions......Page 469
12.12.2 The little Dirac operator......Page 471
12.13 The U(1) Problem Revisited......Page 475
12.14 Conclusion......Page 480
13.1 Introduction......Page 484
13.2 Heat kernel and Seeley expansion......Page 485
13.3 The Atiyah-Singer Index theorem......Page 489
13.4 Fermions in an Instanton potential......Page 491
13.5 Chiral condensate and symmetry breaking......Page 496
13.6 Polyakov loop-operator and screening......Page 504
13.7 Conclusion......Page 507
14.1 Introduction......Page 510
14.2.1 Consistent anomaly......Page 515
14.2.2 More about cocycles......Page 519
14.2.3 Gauss anomaly......Page 521
14.2.4 Relation between consistent and covariant anomaly......Page 522
14.3 Isomorphic Representations of Chiral QCD2......Page 525
14.3.1 Gauge-invariant embedding......Page 526
14.3.2 External Field Ward Identities......Page 528
14.3.3 Construction of the one-Cocycle from the Anomaly......Page 534
14.3.4 Bosonic Action in the GNI and GI Formulation......Page 535
14.3.5 Symmetries of the Model......Page 539
14.3.6 Relation of Source Currents in GNI and GI Formulations......Page 541
14.3.7 Poisson Algebra of the Currents......Page 542
14.3.8 Hamiltonian Quantization......Page 546
14.3.9 Fermionization of a1[A g]......Page 555
14.3.10 BRST Quantization of GI Formulation......Page 556
14.3.11 Chiral QCD2 in Terms of Chiral Bosons......Page 563
14.4 Constraint Structure from the Fermionic Hamiltonian......Page 568
14.5.1 Gauge non-invariant formulation......Page 576
14.5.2 Gauge-invariant formulation......Page 585
14.6 Conclusion......Page 588
15.1 Introduction......Page 594
15.2 The JR Model......Page 595
15.3.1 Hamiltonian and constraints......Page 597
15.3.2 Commutation relations......Page 599
15.3.3 Current-potential and bosonic representation of fermion field......Page 601
15.3.4 Energy-momentum tensor......Page 603
15.3.5 Vector-field two-point function......Page 604
15.4.1 Hamiltonian and constraints......Page 605
15.4.2 Implementation of gauge conditions......Page 607
15.4.3 Isomorphism between GI and GNI formulations: phase space view......Page 609
15.4.4 WZ term and BFT Hamiltonian embedding......Page 612
15.4.5 Alternative approach to quantization......Page 617
15.4.6 Operator solution in Lorentz-type gauges......Page 618
15.5 Path-Integral Formulation......Page 619
15.6.1 Perturbative analysis in the GNI formulation......Page 626
15.6.2 Perturbative analysis in the GI formulation......Page 631
15.7 Anomalous Poisson Brackets Revisited......Page 633
15.7.1 Operator view of anomalous Poisson brackets......Page 634
15.7.3 Reconstruction of commutators of the GNI formulation......Page 636
15.8 Chiral QED2 in terms of Chiral Bosons......Page 639
15.9 Conclusion......Page 642
16.1 Introduction......Page 646
16.2 Conformal transformations and conformal group......Page 647
16.2.2 The conformal group in D dimensions......Page 648
16.3 The conformal group in two dimensions......Page 654
16.3.1 Mobius transformations......Page 656
16.4.1 Primary and quasi-primary fields......Page 660
16.4.2 Radial quantization......Page 667
16.4.3 Descendants of primary fields......Page 672
16.4.4 Virasoro algebra......Page 676
16.5 Realization of Conformal Algebra for c < 1......Page 684
16.6 Superconformal Symmetry......Page 689
16.7 Conclusion......Page 692
17.2 Conformal algebra and Ward identities......Page 696
17.3.1 The Wess-Zumino-Witten field......Page 701
17.3.2 The non-Abelian Thirring field at the Critical Point......Page 707
17.4 Coset description of CQFT......Page 712
17.4.1 Coset realization of the FQS minimal unitary series......Page 713
17.4.2 Fermionic coset realization of SU(N)1......Page 714
17.4.3 Fermionic coset realization of FQS series......Page 717
17.4.4 Reduction formula for negative level WZW fields......Page 719
17.5.1 Fermionic coset description of the critical Ising model......Page 723
17.6 Conclusions......Page 731
18.1 Introduction......Page 734
18.2 The Nambu-Goto string......Page 735
18.3.1 Uniqueness of the Polyakov action......Page 737
18.3.2 Quantum Gravity......Page 739
18.4 The Liouville theory......Page 747
18.4.1 The classical Liouville theory......Page 748
18.4.2 The quantum Liouville theory......Page 751
18.5.1 Canonical quantization and SL(2 R) symmetry......Page 754
18.5.2 Operator product expansions and Ward identities......Page 760
18.5.3 Interaction of matter fields with gravity......Page 761
18.5.4 Two-Dimensional Supergravity......Page 763
18.6 Conclusion......Page 769
19 Final Remarks......Page 772
A Notation (Minkowski Space)......Page 776
B Notation (Euclidean Space)......Page 782
C Further Conventions......Page 786
D Functional Bosonization of the Massive Thirring Model......Page 790
E Bosonization of the Fermionic Kinetic Term......Page 794
F Classical Integrability in the Massive Thirring Model......Page 796
G Quantum Non-Local Charge: Action on Asymptotic States......Page 798
H S-Matrices......Page 802
I Complete S-matrix of the Gross-Neveu Model......Page 806
J Poisson Brackets and Commutators......Page 810
K Chiral Bosons......Page 812
L Axial Anomaly from Dispersion Relations......Page 818
M Loop Expansion in QCD2......Page 822
Index......Page 827