Affine Lie algebras and vertex tensor categories

In this paper, we apply the general theory of tensor products of modules for a vertex operator algebra developed in [HL1]–[HL6] and [H1]–[H2] to the case of the Wess-Zumino-Novikov-Witten models (WZNW models) and related models in conformal field theory. Together with these papers, this paper, among other things, completes the solution of the open problem of constructing the desired braided tensor category structure on the category of finite direct sums of standard (integrable highest weight) modules of a fixed positive integral level k for an affine Lie algebra ĝ. Here we call this category, which is particularly important from the viewpoint of conformal field theory and related mathematics, the category generated by the standard ĝ-modules of level k ∈ Z+. In [MS], Moore and Seiberg discovered among other things that if one assumes that the standard ĝ-modules of level k give a rational conformal field theory, then the category generated by these standard modules has a natural braided tensor category structure (as defined in [JS]). In fact, the main assumption needed in the genus-zero part of their work is the existence of a suitable operator product expansion of chiral vertex operators. This is essentially equivalent to assuming the associativity of intertwining operators, in the language of vertex operator algebra theory. The work [MS] being focused on the problem of classification and not construction of conformal field theories, there was no attempt in [MS] to prove the existence of the operator product expansion of chiral vertex operators or to construct conformal field theories associated to ĝ and k. As far as we know, the associativity of intertwining operators (or the existence of the operator