Particles and Fields

The focus of this volume is on quantum field theory: inegrable theories, statistical systems, and applications to condensed-matter physics. It covers some of the most significant recent advances in theoretical physics at a level accessible to advanced graduate students. The contributions, each by a noted researcher, dicuss such topics as: some remarkable features of integrable Toda field theories (E. Corrigan), properties of a gas of interacting Fermions in a lattice of magnetic ions (J. Feldman &. al.), how quantum groups arise in three-dimensional topological quantum field thory (D. Freed), a method for computing correlation functions of solvable lattice models (T. Miwa), matrix models discussed from the point of view of integrable systems (A. Morozov), localization of path integrals in certain equivariant cohomologies (A. Niemi), Calogero-Moser systems (S. Ruijsenaars), planar gauge theories with broken symmetries (M. de Wild Propitius & F.A. Bais), quantum-Hall fluids (A. Capelli & al.), spectral theory of quantum vortex operators (P.I. Ettinghoff).

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.