iv : h ep - t h / 93 12 06 5 v 1 9 D ec 1 99 3 Vertex Operator Superalgebras and Their Representations ∗ †

Vertex operator algebras (VOA) were introduced in physics by Belavin, Polyakov and Zamolodchikov [BPZ] and in mathematics by Borcherds [B]. For a detailed exposition of the theory of VOAs see [FLM] and [FHL]. In a remarkable development of the theory, Zhu [Z] constructed an associative algebra A(V ) corresponding to a VOA V and established a 1-1 correspondence between the irreducible representations of V and those of A(V ). Furthermore, Frenkel and Zhu [FZ] defined an A(V )-module A(M) for any V -module M and then described the fusion rules in terms of the modules A(M). An important feature of these constructions is that A(V ) and A(M) can usually be computed explicitly. For example, they enabled Frenkel and Zhu to prove the rationality and compute the fusion rules of 1991 Mathematics Subject Classification. Primary 17b65; Secondary 17A70, 17B67. This paper is in the final form, and no version of it will be submitted for publication elsewhere.