Generalized Rationality and a “Jacobi Identity” for Intertwining Operator Algebras

We prove a generalized rationality property and a new identity that we call the “Jacobi identity” for intertwining operator algebras. Most of the main properties of genus-zero conformal field theories, including the main properties of vertex operator algebras, modules, intertwining operators, Verlinde algebras, and fusing and braiding matrices, are incorporated into this identity. Together with associativity and commutativity for intertwining operators proved by the author in [H4] and [H6], the results of the present paper solve completely the problem of finding a natural purely algebraic structure on the direct sum of all inequivalent irreducible modules for a suitable vertex operator algebra. Two equivalent definitions of intertwining operator algebra in terms of this Jacobi identity are given. 0 Introduction This paper is devoted to a study of certain general features of representations of a vertex operator algebra. The basic definitions that we shall need are recalled in Section 1 below. The direct sum of all inequivalent irreducible modules for a suitable vertex operator algebra, equipped with intertwining operators, has a natural algebraic structure called intertwining operator algebra (see [H4] and [H7]). Intertwining operator algebras are multivalued analogues of vertex operator algebras. They were first defined using the convergence property, associativity and skew-symmetry as the main axiom. It is natural to ask whether