Particles and Fields

The present volume has its source in the CAP-CRM summer school on "Particles and Fields" that was held in Banff in the summer of 1994. Over the years, the Division of Theoretical Physics of the Canadian Associa tion of Physicists (CAP) has regularly sponsored such schools on various theoretical and experimental topics. In 1994, the Centre de Recherches Mathematiques (CRM) lent its support to the event. This institute, located in Montreal, is one of Canada's national research centers in the mathe matical sciences. Its mandate includes the organization of scientific events across Canada and since 1994 the CRM has been holding a yearly summer school in Banff as part of its thematic program. The summer school, whose lectures are collected here, has thus become a tradition. The focus of the school was integrable theories, matrix models, statistical systems, field theory and its applications to condensed matter physics, as well as certain aspects of algebra, geometry, and topology. This covers some of the most significant advances in modern theoretical physics. The present volume updates and expands these lectures and reflects the high pedagogical level of the school. The first chapter by E. Corrigan describes some of the remarkable fea tures of the integrable Toda field theories which are associated with affine Dynkin diagrams. The second chapter by J. Feldman, H. Knorrer, D. Leh mann, and E.

This volume focuses 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.

1 Recent Developments in Affine Toda Quantum Field Theory.- 1 Introduction.- 2 Classical Integrability and Classical Data.- 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.- 3 A Single-Slice Fermionic Cluster Expansion.- 4 References.- 3 Quantum Groups from Path Integrals.- 1 Classical Field Theory.- 2 Categories, Finite Groups, and Covering Spaces.- 3 Generalized Path Integrals.- 4 The Quantum Group.- 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.- 3 Generalized Kontsevich Model.- 4 Kp/Toda ?-Function in Terms of Free Fermions.- 5 ?-Function as a Group-Theoretical Quantity.- 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 NonrelativisticCalogero-Moser and Toda Systems.- 3 Relativistic Versions at the Classical Level.- 4 Quantum Calogero-Moser and Toda Systems.- 5 Action-Angle Transforms.- 6 Eigenfunction Transforms.- 7 References.- 8 Discrete Gauge Theories.- 1 Broken Symmetry Revisited.- 2 Basics.- 3 Algebraic Structure.- 4 $${overline D _2}$$ Gauge Theory.- 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.- 3 Existing Theories of Edge Excitations and Experiments.- 4W1+? Minimal Models.- 5 Further Developments.- 6 References.- 10 On the Spectral Theory of Quantum Vertex Operators 469 Pavel I. Etingof.- 1 Basic Definitions.- 2 Spectral Properties of Vertex Operators.- 3 The Semi-Infinite Tensor Product Construction.- 4 Computation of the Leading Eigenvalue and Eigenvector.- 5 References.