Introduction

Abstract

The notion of vertex algebra, introduced by Borcherds [Bo] in a mathematical setting, and by Prenkel, Lepowsky and Meurman [FLM2] in the modified form, “vertex operator algebra,” that we will need to use in the present work, arose naturally in Frenkel, Lepowsky and Meurman’s vertex operator construction [FLM1] [FLM2] of the Monster sporadic finite simple group [Gr]. In fact, they realized the Monster as the symmetry group of a special vertex operator algebra, the “moonshine module.” (See the introduction of [FLM2] for a historical discussion, including the important contribution of Borcherds’ announcement [Bo].) Meanwhile, conformai field theory, a physical theory whose algebraic structure was developed systematically by Belavin, Polyakov and Zamolodchikov [BPZ] in a physical context, has been becoming more and more attractive to mathematicians. In the study of conformal field theories, physicists also arrived at, though perhaps without a completely rigorous definition, the notion of “chiral algebra,” a notion which essentially coincides with the notion of vertex operator algebra (see e.g. [MS]).