# Representation theory and tensor product theory for vertex operator algebras

## TL;DR

This paper develops a comprehensive tensor product theory for modules over vertex operator algebras, establishing foundational properties, relating to existing theories, and proving key formulas for fusion rules, with implications for rational models.

## Contribution

It introduces a universal tensor product construction for vertex operator algebra modules, proves its properties, and connects it with existing fusion rule formulas and module theory.

## Key findings

- Tensor product construction satisfies unital and commutativity properties.
- Fusion rules match those from Tsuchiya and Kanie's method for rational models.
- Existence of a unique maximal submodule in generalized modules under certain conditions.

## Abstract

We first formulate a definition of tensor product for two modules for a vertex operator algebra in terms of a certain universal property and then we give a construction of tensor products. We prove the unital property of the adjoint module and the commutativity of tensor products, up to module isomorphism. We relate this tensor product construction with Frenkel and Zhu's $A(M)$-theory. We give a proof of a formula of Frenkel and Zhu for fusion rules. We also give the analogue of the ``Hom''-functor of classical Lie algebra theory for vertex operator algebra theory by introducing a notion of ``generalized intertwining operator.'' We prove that the space of generalized intertwining operators from one module to another for a vertex operator algebra is a generalized module. From this result we derive a general form of Tsuchiya and Kanie's ``nuclear democracy theorem'' for any rational vertex operator algebra. This proves that the fusion rules obtained from our construction of tensor products are the same as the fusion rules obtained by using Tsuchiya and Kanie's method, for both WZW models and minimal models. We prove that if $V$ satisfies certain ``finiteness'' and ``semisimplicity'' conditions, then there exists a unique maximal submodule inside the generalized module. Furthermore, we prove that this maximal submodule is isomorphic to the contragredient module of a certain tensor product module. This gives another construction of tensor product modules and this result turns out to be closely related to Huang and Lepowsky's construction.