1 Recent Developments in Affine Toda Quantum Field Theory.- 1 Introduction.- 2 Classical Integrability and Classical Data.- 2.1 Geometry Associated with the Coxeter Element.- 3 Aspects of the Quantum Field Theory.- 4 Dual Pairs.- 5 A Word on Solitons.- 6 Other Matters.- 7 References.- 2 A Class of Fermi Liquids.- 1 Introduction.- 2 Four-Legged Diagrams.- 2.1 The Particle-Particle Bubble.- 2.2 The Particle-Hole Bubble.- 2.3 Higher-Order Diagrams.- 3 A Single-Slice Fermionic Cluster Expansion.- 4 References.- 3 Quantum Groups from Path Integrals.- 1 Classical Field Theory.- 1.1 Classical Actions.- 1.2 The Wess-Zumino-Witten Action.- 2 Categories, Finite Groups, and Covering Spaces.- 2.1 Going Further.- 2.2 Finite-Gauge Theory.- 3 Generalized Path Integrals.- 3.1 Path Integral Quantization.- 3.2 Beyond Quantum Hilbert Spaces.- 3.3 Quantum Finite-Gauge Theory.- 4 The Quantum Group.- 4.1 The 2-Hilbert Space.- 4.2 Locality and Gluing.- 5 References.- 4 Half Transfer Matrices in Solvable Lattice Models.- 1 The Six-Vertex Model.- 2 The Antiferromagnetic Regime.- 3 Corner Transfer Matrix.- 4 Half Transfer Matrix.- 5 Commutation Relations.- 6 Correlation Functions.- 7 Two-Point Functions.- 8 Discussion.- 9 References.- 5 Matrix Models as Integrable Systems.- 1 Introduction.- 2 The Basic Example: Discrete 1-Matrix Model.- 2.1 Ward Identities.- 2.2 CFT Interpretation of 1-Matrix Model.- 2.3 1-Matrix Model in Eigenvalue Representation.- 2.4 Kontsevich-Like Representation of the 1-Matrix Model.- 3 Generalized Kontsevich Model.- 3.1 Kontsevich Integral. The First Step.- 3.2 Itzykson-Zuber Integral and Duistermaat-Heckmann Theorem.- 3.3 Kontsevich Integral. The Second Step.- 3.4 "Phases" of Kontsevich Integral. GKM as the "Quantum Piece" of $${\mathcal{F}_V}\{ L\} $$ in the Kontsevich Phase.- 3.5 Relation Between Time-and Potential-Dependencies.- 3.6 Kac-Schwarz Problem.- 3.7 Ward Identities for GKM.- 4 Kp/Toda ?-Function in Terms of Free Fermions.- 4.1 Explicit Definition.- 4.2 Basic Determinant Formula for the Free-Fermion Correlator.- 4.3 KP Hierarchy and Other Reductions.- 4.4 Fermion Correlator in Miwa Coordinates.- 4.5 1-Matrix Model versus Toda-Chain Hierarchy.- 5 ?-Function as a Group-Theoretical Quantity.- 5.1 From Intertwining Op0F4erators to Bilinear Equations...- 5.2 The Case of KP/Toda ?-Functions.- 5.3 Example of SL(2) q.- 5.4 Comments on the Quantum Deformation of KP/Toda.- ?-Functions.- 6 Conclusion.- 7 References.- 6 Localization, Equivariant Cohomology, and Integration Formulas 211.- 1 Symplectic Geometry.- 2 Equivariant Cohomology.- 3 Duistermaat-Heckman Integration Formula.- 4 Degeneracies.- 5 Equivariant Characteristic Classes.- 6 Loop Space.- 7 Example: Atiyah-Singer Index Theorem.- 8 Duistermaat-Heckman in Loop Space.- 9 General Integrable Models.- 10 Mathai-Quillen Formalism.- 11 Short Review of Morse Theory.- 12 Equivariant Mathai-Quillen Formalism.- 13 Equivariant Morse Theory.- 14 Loop Space and Morse Theory.- 15 Loop Space and Equivariant Morse Theory.- 16 Poincaré Supersymmetry and Equivariant Cohomology..- 17 References.- 7 Systems of Calogero-Moser Type.- 1 Introduction.- 2 Classical Nonrelativistic Calogero-Moser and Toda Systems.- 2.1 Background: Classical Mechanics/Symplectic Geometry.- 2.2 Calogero-Moser Systems.- 2.3 Toda Systems.- 3 Relativistic Versions at the Classical Level.- 3.1 The Defining Dynamics and its Commuting Integrals...- 3.2 Lax Matrices and Their Interrelationships.- 4 Quantum Calogero-Moser and Toda Systems.- 4.1 Background: Quantum Mechanics/Hilbert Space Theory.- 4.2 The Nonrelativistic Case: Commuting PDOs.- 4.3 The Relativistic Case: Commuting A?Os.- 5 Action-Angle Transforms.- 5.1 Introductory Examples.- 5.2 Wave Maps and Pure Soliton Systems.- 5.3 Systems of Type I, II, and III.- 6 Eigenfunction Transforms.- 6.1 Preliminaries.- 6.2 Type III Eigenfunctions for Arbitrary N.- 6.3 Type II and IV Eigenfunctions for N = 2.- 7 References.- 8 Discrete Gauge Theories.- 1 Broken Symmetry Revisited.- 2 Basics.- 2.1 Introduction.- 2.2 Braid Groups.- 2.3 ?N Gauge Theory.- 2.4 Non-Abelian Discrete Gauge Theories.- 3 Algebraic Structure.- 3.1 Quantum Double.- 3.2 Truncated Braid Groups.- 3.3 Fusion, Spin, Braid Statistics, and All That.- 4 $${\overline D _2}$$ Gauge Theory.- 4.1 Alice in Physics.- 4.2 Scattering Doublet Charges Off Alice Fluxes.- 4.3 Non-Abelian Braid Statistics.- 4.4 Aharonov-Bohm Scattering.- 4.5 B(3,4) and P(3,4).- 5 Concluding Remarks and Outlook.- 6 References.- 9 Quantum Hall Fluids as W1+?
Minimal Models.- 1 Introduction.- 2 Dynamical Symmetry and Kinematics of Incompressible Fluids.- 2.1 Classical Fluids.- 2.2 Quantum Fluids and Their Edge Excitations.- 2.3 Classification of QHE Universality Classes.- 3 Existing Theories of Edge Excitations and Experiments.- 3.1 Hierarchical Trial Wave Functions.- 3.2 The Chiral Boson Theory of the Edge Excitations..- 3.3 The Jain Hierarchy.- 3.4 Experiments.- 4
W1+? Minimal Models.- 4.1 The Theory of
W1+? Representations.- 4.2 The
W1+? Minimal Models.- 4.3 Non-Abelian Fusion Rules and Non-Abelian Statistics.- 4.4 The Degeneracy of Excitations Above the Ground State.- 4.5 Remarks on the SU(m) and $$S\widehat {U(m}{)_1}$$ Symmetries.- 5 Further Developments.- 6 References.- 10 On the Spectral Theory of Quantum Vertex Operators 469 Pavel I. Etingof.- 1 Basic Definitions.- 1.1 Quantum Groups.- 1.2 Representations.- 1.3 Vertex Operators.- 1.4 The Fock Space.- 1.5 Bosonization of $${U_q}(\widehat {\mathfrak{s}{\mathfrak{l}_2}})$$.- 1.6 Bosonization of Vertex Operators.- 1.7 Boson-Fermion Correspondence.- 2 Spectral Properties of Vertex Operators.- 2.1 Vertex Operators as Power Series in q.- 2.2 Composition of Vertex Operators.- 2.3 The Operators F+-(0) and F-+(0).- 2.4 The Highest Eigenvalue of F-+(q), F+-(q).- 3 The Semi-Infinite Tensor Product Construction.- 3.1 The Kyoto Conjecture.- 3.2 The Kyoto Homomorphism.- 4 Computation of the Leading Eigenvalue and Eigenvector.- 5 References.

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