Tensor Products of Modules for a Vertex Operator Algebra and Vertex Tensor Categories Yi-zhi Huang and James Lepowsky

In this paper, we present a theory of tensor products of classes of modules for a vertex operator algebra. We focus on motivating and explaining new structures and results in this theory, rather than on proofs, which are being presented in a series of papers beginning with [HL4] and [HL5]. An announcement has also appeared [HL1]. The theory is based on both the formal-calculus approach to vertex operator algebra theory developed in [FLM2] and [FHL] and the precise geometric interpretation of the notion of vertex operator algebra established in [H1]. Recently, mathematicians have been more and more attracted to conformal field theory, a physical theory which plays an important role in both condensed matter physics and string theory. Much of the research on conformal field theory has been centered on the conformal field theories determined by holomorphic fields of weight 1—the theories associated with certain highest-weight representations of affine KacMoody algebras. Many important structures and concepts which have arisen in recent years are related to this special class of conformal field theories—Wess-ZuminoNovikov-Witten (WZNW) models, the Knizhnik-Zamolodchikov equations and associated monodromy, quantum groups, braided tensor categories, the Jones polynomial and generalizations, three-manifold invariants, Chern-Simons theory, the Verlinde formula, etc. (see for instance [Wi1], [KZ], [J], [K1], [K2], [Ve], [TK], [MS], [Wi2], [TUY], [Dr1], [Dr2], [RT], [SV], [Va], [KL1]–[KL5], [Fi], [Fa]). But there are also other important mathematical structures that can be studied as conformal-field-theoretic structures, in particular, highest weight representations of the Virasoro algebra, Walgebras and their representations, and most particularly for us, the moonshine module V ♮ for the Fischer-Griess Monster sporadic finite simple group [Gr] constructed in [FLM1] and [FLM2]. For the conformal field theories associated with these structures, there are no nonzero holomorphic fields of weight 1, and correspondingly, the special methods for studying those conformal field theories associated with affine Lie algebras do not apply. We must have a broader viewpoint than that of this special class of conformal field theories.