A Proof of the Fusion Rules Theorem

We prove that the space of intertwining operators associated with certain admissible modules over vertex operator algebras is isomorphic to a quotient of the vector space of conformal blocks on a three-pointed rational curve defined by the same data. This provides a new proof and alternative version of Frenkel and Zhu’s fusion rules theorem, in terms of the dimension of certain bimodules over Zhu’s algebra, without the assumption of rationality.