| Introduction | p. 17 |
| Free Fields | p. 25 |
| Introduction | p. 25 |
| Bosonic Free Fields | p. 25 |
| Fermionic Free Fields | p. 29 |
| Bosonization of Massless Fermions | p. 31 |
| The RS-Model | p. 35 |
| Conclusions | p. 38 |
| The Thirring model | p. 41 |
| Introduction | p. 41 |
| The Massless Thirring Model | p. 42 |
| The Massive Thirring Model | p. 45 |
| Equivalence with sine-Gordon equation | p. 45 |
| Classical conservation laws | p. 47 |
| Quantum conservation laws | p. 48 |
| Bosonization Revisited | p. 53 |
| Fermions in terms of bosons | p. 53 |
| The Soliton as a Disorder Parameter | p. 55 |
| Conclusion | p. 61 |
| Determinants and Heat Kernels | p. 65 |
| Introduction | p. 65 |
| Functional Determinant, one-loop diagram | p. 66 |
| Determinants and the Generalized Zeta-Function | p. 69 |
| One Point Compactification | p. 75 |
| The associated Dirac operator | p. 78 |
| Calculating Seeley Coefficients | p. 81 |
| The perturbative approach | p. 81 |
| The Schwinger-DeWitt method | p. 83 |
| The Fujikawa method | p. 86 |
| Computing Functional Determinants | p. 88 |
| [zeta]-function regularization | p. 88 |
| Proper-time regularization | p. 90 |
| The Fujikawa point of view | p. 91 |
| A Theorem on a one parameter family of factorizable operators | p. 95 |
| The QCD[subscript 2] functional determinant | p. 98 |
| Zero-modes | p. 101 |
| Axial anomaly equation in the presence of zero-modes | p. 101 |
| Atiyah-Singer Index Theorem | p. 104 |
| Ambiguities in Functional Determinants | p. 107 |
| Ambiguities in the regularization | p. 107 |
| Dependence on the scale parameter | p. 108 |
| Mass expansion in proper-time regularization | p. 111 |
| The Finite Temperature Heat Kernel | p. 115 |
| Scalar field in a static background potential | p. 117 |
| Scalar field in a static background gauge potential | p. 119 |
| Conclusion | p. 123 |
| Self-Interacting fermionic models | p. 127 |
| Introduction | p. 127 |
| The O(N) Invariant Gross--Neveu Model | p. 127 |
| Classical conservation laws | p. 128 |
| Effective potential and [beta]-function in a 1/N expansion | p. 129 |
| The 1/N Expansion: Feynman rules | p. 133 |
| Leading order S-matrix elements | p. 135 |
| Quantization of the non-local charge | p. 138 |
| Chiral Gross--Neveu Model | p. 141 |
| Cancellation of infrared singularities | p. 142 |
| The 1/N expansion | p. 144 |
| Operator formulation | p. 146 |
| Quantization of non-local charge | p. 151 |
| Conclusions and Physical Interpretation | p. 152 |
| Non-linear [sigma] Models: Classical Aspects | p. 155 |
| Historical development | p. 155 |
| Sigma models and current algebra | p. 156 |
| Two-dimensional [sigma] models: preliminaries | p. 158 |
| Purely Bosonic Non-linear [sigma] Models | p. 166 |
| Formal developments | p. 166 |
| Dual symmetry and higher conservation laws | p. 172 |
| An explicit example: the Grassmannians | p. 182 |
| Non-linear [sigma] Models with Fermions | p. 184 |
| Definition and properties | p. 184 |
| Dual symmetry and higher conservation laws | p. 188 |
| Construction of an explicit example | p. 195 |
| Analogies with 4D Gauge Theories | p. 200 |
| Concluding Remarks | p. 204 |
| Non-linear [sigma] Models - Quantum Aspects | p. 211 |
| Introduction | p. 211 |
| Grassmannian Bosonic Models | p. 212 |
| 1/N expansion | p. 212 |
| Renormalization | p. 218 |
| Infrared divergencies | p. 219 |
| Physical interpretation of the results | p. 221 |
| Grassmannian Models and Fermions | p. 222 |
| 1/N expansion and Feynman rules | p. 222 |
| Physical interpretation of the results | p. 227 |
| Quantization of Higher Conservation Laws | p. 233 |
| Purely bosonic sigma models and anomalies | p. 233 |
| Fermionic interaction and anomaly cancellation | p. 239 |
| Algebra of non-local charges | p. 242 |
| Bosonic O(N)-symmetric sigma models | p. 242 |
| Non-local charges in the WZNW model | p. 254 |
| Perturbative Renormalization | p. 257 |
| Background Field Method | p. 257 |
| Parallelizable manifolds; applications to string theory | p. 263 |
| Anomalous Non-Linear [sigma] Models in four dimensions | p. 265 |
| Conclusion | p. 266 |
| Exact S-matrices of 2D Models | p. 273 |
| Introduction | p. 273 |
| Consequences of higher conservation laws | p. 273 |
| Factorizable S-matrix | p. 274 |
| Fusion rules | p. 278 |
| Bound state scattering | p. 280 |
| S-matrices and Conservation Laws | p. 280 |
| SU(N) invariant S-matrices | p. 280 |
| Sine-Gordon and massive Thirring models | p. 282 |
| Exact S-matrix for O(N) symmetry | p. 287 |
| The Z[subscript N] invariant S-matrix | p. 288 |
| Quantum Non-Local Charges and S-Matrices | p. 289 |
| S-matrices of purely fermionic models | p. 289 |
| S-matrices of non-linear sigma models | p. 293 |
| Boundary S-matrices | p. 303 |
| Further Developments | p. 307 |
| Conclusion | p. 308 |
| The Wess--Zumino--Witten Theory | p. 313 |
| Introduction | p. 313 |
| Existence of a Critical Point | p. 315 |
| Properties at the Critical Point | p. 318 |
| The Polyakov--Wiegmann formula | p. 319 |
| The Affine algebra | p. 320 |
| The WZW fields in terms of fermions | p. 322 |
| The Sugawara form of the energy-momentum tensor | p. 323 |
| The non-Abelian bosonization in the operator language | p. 324 |
| Properties off the Critical Point | p. 325 |
| Integrability of the WZNW action | p. 326 |
| On the solution off the critical point | p. 327 |
| Supersymmetric W ZW model | p. 329 |
| Conclusion | p. 331 |
| QED[subscript 2]: Operator Approach | p. 333 |
| Introduction | p. 333 |
| The Massless Schwinger Model | p. 335 |
| Quantum solution | p. 335 |
| The Maxwell current | p. 337 |
| Chiral densities | p. 340 |
| Vacuum structure | p. 341 |
| Gauge transformations | p. 345 |
| Correlation functions and violation of clustering | p. 348 |
| Absence of charged states (screening) | p. 349 |
| The quark-antiquark potential | p. 351 |
| Adding flavour | p. 353 |
| Fractional winding number and the U(1) problem | p. 356 |
| The Massive Schwinger Model | p. 360 |
| Equivalent bosonic formulation | p. 360 |
| The quantum Dirac equation | p. 362 |
| Vacuum structure and all that | p. 365 |
| Screening versus confinement | p. 366 |
| Adding flavour | p. 374 |
| Lorentz transformation properties | p. 380 |
| The MSM as the limit of a massive vector theory | p. 383 |
| Conclusion | p. 386 |
| Quantum Chromodynamics | p. 391 |
| Introduction | p. 391 |
| The 1/N expansion: 't Hooft model | p. 394 |
| Currents, Green functions and determinants | p. 399 |
| Tree graph expansion of the current | p. 400 |
| Recovering the QCD[subscript 2] effective action | p. 402 |
| Fermion Green Function | p. 405 |
| Local decoupled formulation and BRST constraints | p. 408 |
| Local decoupled partition function and BRST symmetries | p. 409 |
| Systematic derivation of the constraints | p. 414 |
| Non-local decoupled formulation and BRST constraints | p. 417 |
| Non-local decoupled partition function and BRST symmetries | p. 417 |
| The physical Hilbert space | p. 421 |
| The QCD[subscript 2] vacuum | p. 422 |
| Massive two-dimensional QCD | p. 425 |
| Screening in two-dimensional QCD | p. 427 |
| Further algebraic aspects | p. 433 |
| Conclusions | p. 434 |
| QED[subscript 2]: Functional Approach | p. 439 |
| Introduction | p. 439 |
| Equivalent Bosonic Action | p. 440 |
| Gauge Invariant Correlation Functions | p. 441 |
| The external field current, and chiral densities | p. 441 |
| Vacuum Structure | p. 442 |
| Chirality of the vacuum | p. 443 |
| Why Study Gauge-Invariant Correlators | p. 447 |
| Screening versus Confinement | p. 448 |
| Quasi-Periodic Boundary Conditions and the [theta]-Vacuum | p. 450 |
| Axial anomaly and the Dirac sea | p. 454 |
| Functional Representation of Tunneling Amplitudes | p. 456 |
| Interpretation of the Result | p. 458 |
| Zero modes | p. 460 |
| Calculation of det i D from the anomaly equation | p. 462 |
| Eigenvalue Spectrum of the Dirac Operator | p. 464 |
| Zero Modes and Boundary-Value Problem | p. 467 |
| Free Dirac operator and non-local boundary conditions | p. 468 |
| The little Dirac operator | p. 470 |
| The U (1) Problem Revisited | p. 474 |
| Conclusion | p. 479 |
| The Finite Temperature Schwinger Model | p. 483 |
| Introduction | p. 483 |
| Heat kernel and Seeley expansion | p. 484 |
| The Atiyah-Singer Index theorem | p. 488 |
| Fermions in an Instanton potential | p. 490 |
| Chiral condensate and symmetry breaking | p. 495 |
| Polyakov loop-operator and screening | p. 503 |
| Conclusion | p. 506 |
| Non-Abelian Chiral Gauge Theories | p. 509 |
| Introduction | p. 509 |
| Anomalies and Cocycles | p. 514 |
| Consistent anomaly | p. 514 |
| More about cocycles | p. 518 |
| Gauss anomaly | p. 520 |
| Relation between consistent and covariant anomaly | p. 521 |
| Isomorphic Representations of Chiral QCD[subscript 2] | p. 524 |
| Gauge-invariant embedding | p. 525 |
| External Field Ward Identities | p. 527 |
| Construction of the one-Cocycle from the Anomaly | p. 533 |
| Bosonic Action in the GNI and GI Formulation | p. 534 |
| Symmetries of the Model | p. 538 |
| Relation of Source Currents in GNI and GI Formulations | p. 540 |
| Poisson Algebra of the Currents | p. 541 |
| Hamiltonian Quantization | p. 545 |
| Fermionization of [alpha subscript 1] [A, g] | p. 554 |
| BRST Quantization of GI Formulation | p. 555 |
| Chiral QCD[subscript 2] in Terms of Chiral Bosons | p. 562 |
| Constraint Structure from the Fermionic Hamiltonian | p. 567 |
| Chiral QCD[subscript 2] in the local decoupled formulation | p. 575 |
| Gauge non-invariant formulation | p. 575 |
| Gauge-invariant formulation | p. 584 |
| Conclusion | p. 587 |
| Chiral Quantum Electrodynamics | p. 593 |
| Introduction | p. 593 |
| The JR Model | p. 594 |
| Quantization in the GNI Formulation | p. 596 |
| Hamiltonian and constraints | p. 596 |
| Commutation relations | p. 598 |
| Current-potential and bosonic representation of fermion field | p. 600 |
| Energy-momentum tensor | p. 602 |
| Vector-field two-point function | p. 603 |
| Fermionic two-point function | p. 604 |
| Quantization in the GI Formulation | p. 604 |
| Hamiltonian and constraints | p. 604 |
| Implementation of gauge conditions | p. 606 |
| Isomorphism between GI and GNI formulations: phase space view | p. 608 |
| WZ term and BFT Hamiltonian embedding | p. 611 |
| Alternative approach to quantization | p. 616 |
| Operator solution in Lorentz-type gauges | p. 617 |
| Path-Integral Formulation | p. 618 |
| Perturbative Analysis in the Fermionic Formulation | p. 625 |
| Perturbative analysis in the GNI formulation | p. 625 |
| Perturbative analysis in the GI formulation | p. 630 |
| Anomalous Poisson Brackets Revisited | p. 632 |
| Operator view of anomalous Poisson brackets | p. 633 |
| Bjorken-Johnson-Low view of anomalous Poisson brackets | p. 635 |
| Reconstruction of commutators of the GNI formulation | p. 635 |
| Chiral QED[subscript 2] in terms of Chiral Bosons | p. 638 |
| Conclusion | p. 641 |
| Conformally Invariant Field Theory | p. 645 |
| Introduction | p. 645 |
| Conformal transformations and conformal group | p. 646 |
| Dilatations | p. 647 |
| The conformal group in D dimensions | p. 647 |
| The conformal group in two dimensions | p. 653 |
| Mobius transformations | p. 655 |
| The BPZ construction | p. 659 |
| Primary and quasi-primary fields | p. 659 |
| Radial quantization | p. 666 |
| Descendants of primary fields | p. 671 |
| Virasoro algebra | p. 675 |
| Realization of Conformal Algebra for c [ 1 | p. 683 |
| Superconformal Symmetry | p. 688 |
| Conclusion | p. 691 |
| Conformal Field Theory with Internal Symmetry | p. 695 |
| Introduction | p. 695 |
| Conformal algebra and Ward identities | p. 695 |
| Realizations of non-Abelian conformal algebra | p. 700 |
| The Wess-Zumino-Witten field | p. 700 |
| The non-Abelian Thirring field at the Critical Point | p. 706 |
| Coset description of CQFT | p. 711 |
| Coset realization of the FQS minimal unitary series | p. 712 |
| Fermionic coset realization of SU (N)[subscript 1] | p. 713 |
| Fermionic coset realization of FQS series | p. 716 |
| Reduction formula for negative level WZW fields | p. 718 |
| Critical statistical models | p. 722 |
| Fermionic coset description of the critical Ising model | p. 722 |
| Conclusions | p. 730 |
| 2D gravity and string related topics | p. 733 |
| Introduction | p. 733 |
| The Nambu-Goto string | p. 734 |
| The effective action of 2D quantum gravity | p. 736 |
| Uniqueness of the Polyakov action | p. 736 |
| Quantum Gravity | p. 738 |
| The Liouville theory | p. 746 |
| The classical Liouville theory | p. 747 |
| The quantum Liouville theory | p. 750 |
| Gravity in the light-cone gauge | p. 753 |
| Canonical quantization and SL (2, R) symmetry | p. 753 |
| Operator product expansions and Ward identities | p. 759 |
| Interaction of matter fields with gravity | p. 760 |
| Two-Dimensional Supergravity | p. 762 |
| Conclusion | p. 768 |
| Final Remarks | p. 771 |
| Appendices | |
| Notation (Minkowski Space) | p. 775 |
| Notation (Euclidean Space) | p. 781 |
| Further Conventions | p. 785 |
| Functional Bosonization of the Massive Thirring Model | p. 789 |
| Bosonization of the Fermionic Kinetic Term | p. 793 |
| Classical Integrability in the Massive Thirring Model | p. 795 |
| Quantum Non-Local Charge: Action on Asymptotic States | p. 797 |
| S-Matrices | p. 801 |
| Complete S-matrix of the Gross-Neveu Model | p. 805 |
| Poisson Brackets and Commutators | p. 809 |
| Chiral Bosons | p. 811 |
| Axial Anomaly from Dispersion Relations | p. 817 |
| Loop Expansion in QCD[subscript 2] | p. 821 |
| Index | p. 826 |
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