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.

Inhalt

String models.- to string models and vertex operators.- An introduction to Polyakov's string model.- Conformally invariant field theories in two dimensions.- Lie algebra representations.- Algebras, lattices and strings.- L-algebras and the Rogers-Ramanujan identities.- Structure of the standard modules for the affine Lie algebra A1(1) in the homogeneous picture.- Standard representations of some affine Lie algebras.- Some applications of vertex operators to Kac-Moody algebras.- On a duality of branching coefficients.- The Monster.- A brief introduction to the finite simple groups.- A Moonshine Module for the Monster.- Integrable systems.- Monodromy, solitons and infinite dimensional Lie algebras.- The Riemann-Hilbert decomposition and the KP hierarchy.- Supersymmetric Yang-Mills fields as an integrable system and connections with other non-linear systems.- Lax pairs, Riemann-Hilbert transforms and affine algebras for hidden symmetries in certain nonlinear field theories.- Massive Kaluza-Klein theories and bound states in Yang-Mills.- Local charge algebras in quantum chiral models and gauge theories.- Supergeometry and Kac-Moody algebras.- The Virasoro algebra.- A proof of the no-ghost theorem using the Kac determinant.- Conformal invariance, unitarity and two dimensional critical exponents.- Vacuum vector representations of the Virasoro algebra.- Classical invariant theory and the Virasoro algebra.