Vertex Operator Algebras: The Axiomatic Basics

In this chapter we begin presenting the axiomatic theory of vertex operator algebras. The notion of vertex algebra was introduced in [B1] and, a variant of the notion, that of vertex operator algebra, was developed in [FLM6] and, [FHL]. The notion of vertex operator algebra is the mathematical counterpart of the notion of “operator algebra,” or “chiral algebra,” in conformai field theory, as formalized in [BPZ], and, many of the algebraic considerations in this chapter were introduced and, exploited in more physical language and, settings in the vast physics literature on conformai field theory and, string theory. Our treatment is based on [B1], [FLM6], [FHL], [DL3] and, [Li3]. In Section 3.3 we include a discussion of the relations between the mathematical treatment of “associativity” presented here and, operator product expansions in the sense of conformai field theory and, string theory.