Introduction

Abstract

The rapidly-evolving theory of vertex operator algebras provides deep insight into many important algebraic structures, including the Monster finite simple group and highest weight modules for affine Lie algebras and for the Virasoro algebra. The original motivation for the introduction of the precise notion of vertex operator algebra arose from the problem of realizing the Monster as a symmetry group of a natural infinite-dimensional structure, and in the book [FLM3], the Monster was in fact realized as the automorphism group of a certain vertex operator algebra. Partly motivated by the announcement [FLM1], Borcherds introduced a notion of “vertex algebra” in his announcement [Bor], and the variant of this notion that we call “vertex operator algebra,” equipped with the fundamental “Jacobi identity” as the main axiom, was introduced in [FLM3] and [FHL]. (The reader may wish to consult the introductory material in these works for historical background.) Vertex operator algebras can be viewed as “complex analogues” of both Lie algebras and associative algebras. In the physics literature, a physical counterpart of the notion of vertex operator algebra — the notion of what is now usually called “chiral algebra” — evolved in relation to dual resonance theory and more recently, to string theory and conformal field theory; see for instance for the historical comments in [FLM3] and [MS].