Vertex Operators in Mathematics and Physics

1. Front Matter

   Pages i-xiv

   PDF

2. Introduction

   1. Introduction

      • James Lepowsky

      Pages 1-13

3. String models

   1. Introduction to String Models and Vertex Operators

      • S. Mandelstam

      Pages 15-35

   2. An Introduction to Polyakov’s String Model

      • Orlando Alvarez

      Pages 37-47

   3. Conformally Invariant Field Theories in Two Dimensions

      • Thomas Curtright

      Pages 49-50

4. Lie algebra representations

   1. Algebras, Lattices and Strings

      • P. Goddard, D. Olive

      Pages 51-96

   2. Z-Algebras and the Rogers-Ramanujan Identities

      • James Lepowsky, Robert Lee Wilson

      Pages 97-142

   3. Structure of the Standard Modules for the Affine Lie Algebra A1 (1) in the Homogeneous Picture

      • J. Lepowsky, M. Primc

      Pages 143-162

   4. Standard Representations of Some Affine Lie Algebras

      • Kailash C. Misra

      Pages 163-183

   5. Some Applications of Vertex Operators to Kac-Moody Algebras

      • Alex J. Feingold

      Pages 185-206

   6. On a Duality of Branching Coefficients

      • Michio Jimbo, Tetsuji Miwa

      Pages 207-216

5. The Monster

   1. A Brief Introduction to the Finite Simple Groups

      • Robert L. Griess Jr.

      Pages 217-229

   2. A Moonshine Module for the Monster

      • Igor B. Frenkel, James Lepowsky, Arne Meurman

      Pages 231-273

6. Integrable Systems

   1. Monodromy, Solitons and Infinite Dimensional Lie Algebras

      • Michio Jimbo, Tetsuji Miwa

      Pages 275-290

   2. The Riemann-Hilbert Decomposition and the KP Hierarchy

      • Kimio Ueno

      Pages 291-302

   3. Supersymmetric Yang-Mills Fields as an Integrable System and Connections with Other Non-Linear Systems

      • Ling-Lie Chau

      Pages 303-328

   4. Lax Pairs, Riemann-Hilbert Transforms and Affine Algebras for Hidden Symmetries in Certain Nonlinear Field Theories

      • Yong-Shi Wu, Mo-Lin Ge

      Pages 329-352

   5. Massive Kaluza-Klein Theories and Bound States in Yang-Mills

      • L. Dolan

      Pages 353-372

   6. Local Charge Algebras in Quantum Chiral Models and Gauge Theories

      • Itzhak Bars

      Pages 373-391

   7. Supergeometry and Kac-Moody Algebras

      • B. Julia

      Pages 393-409

James Lepowsky t The search for symmetry in nature has for a long time provided representation theory with perhaps its chief motivation. According to the standard approach of Lie theory, one looks for infinitesimal symmetry -- Lie algebras of operators or concrete realizations of abstract Lie algebras. A central theme in this volume is the construction of affine Lie algebras using formal differential operators called vertex operators, which originally appeared in the dual-string theory. Since the precise description of vertex operators, in both mathematical and physical settings, requires a fair amount of notation, we do not attempt it in this introduction. Instead we refer the reader to the papers of Mandelstam, Goddard-Olive, Lepowsky-Wilson and Frenkel-Lepowsky-Meurman. We have tried to maintain consistency of terminology and to some extent notation in the articles herein. To help the reader we shall review some of the terminology. We also thought it might be useful to supplement an earlier fairly detailed exposition of ours [37] with a brief historical account of vertex operators in mathematics and their connection with affine algebras. Since we were involved in the development of the subject, the reader should be advised that what follows reflects our own understanding. For another view, see [29].1 t Partially supported by the National Science Foundation through the Mathematical Sciences Research Institute and NSF Grant MCS 83-01664. 1 We would like to thank Igor Frenkel for his valuable comments on the first draft of this introduction.

• Lie algebra
• Operator (Math.)
• Operators
• Physics
• Scheitel (Math.)

• Department of Mathematics, Rutgers University, New Brunswick, USA

  J. Lepowsky

• Department of Physics, University of California, Berkeley, USA

  S. Mandelstam

• Department of Mathematics, University of California, Berkeley, USA

  I. M. Singer

• Mathematical Sciences Research Institute, Berkeley, USA

  I. M. Singer

Policies and ethics