A proof of the fusion rules theorem

TL;DR

This paper provides a new proof of the fusion rules theorem for vertex operator algebras, establishing an isomorphism between intertwining operators and conformal blocks without requiring rationality.

Contribution

It offers an alternative proof of Frenkel and Zhu's fusion rules theorem using bimodule dimensions, removing the rationality assumption.

Findings

Proves the isomorphism between intertwining operators and conformal blocks.

Extends the fusion rules theorem to non-rational vertex operator algebras.

Provides a new perspective on the structure of modules over Zhu's algebra.

Abstract

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.