Fusion rules of chiral algebras

TL;DR

This paper generalizes the fusion rules of chiral algebras, defining a tensor product for conformal field theories that is associative and symmetric, and applies it to various algebra cases for calculating fusion restrictions.

Contribution

It introduces a general tensor product framework for chiral algebras, extending previous models and providing a method to compute fusion rule restrictions.

Findings

Tensor product of conformal field theories is associative and symmetric.

Framework applies to W3-algebra and N=1,2 superconformal algebras.

Facilitates calculation of fusion rule restrictions.

Abstract

Recently (hep-th/9307183) we showed that for the case of the WZW- and the minimal models fusion can be understood as a certain ring-like tensor product of the symmetry algebra. In this paper we generalize this analysis to arbitrary chiral algebras. We define the tensor product of conformal field theory in the general case and prove that it is associative and symmetric up to equivalence. We also determine explicitly the action of the chiral algebra on this tensor product. In the second part of the paper we demonstrate that this framework provides a powerful tool for calculating restrictions for the fusion rules of chiral algebras. We exhibit this for the case of the $W_{3}$-algebra and the $N=1$ and $N=2$ NS superconformal algebras.