Generating-function method for fusion rules

TL;DR

This paper develops a generating-function approach to compute fusion rules in affine Lie algebras, extending previous tensor product methods with new inequalities and explicit examples for sp(4) and su(4).

Contribution

It introduces a novel method for deriving fusion rules using a fusion basis and inequalities, improving upon the Kac-Walton algorithm with explicit formulas and threshold levels.

Findings

Derived fusion basis from elementary couplings and inequalities.

Constructed new generating functions for sp(4) and su(4).

Provided closed-form expressions for threshold levels.

Abstract

This is the second of two articles devoted to an exposition of the generating-function method for computing fusion rules in affine Lie algebras. The present paper focuses on fusion rules, using the machinery developed for tensor products in the companion article. Although the Kac-Walton algorithm provides a method for constructing a fusion generating function from the corresponding tensor-product generating function, we describe a more powerful approach which starts by first defining the set of fusion elementary couplings from a natural extension of the set of tensor-product elementary couplings. A set of inequalities involving the level are derived from this set using Farkas' lemma. These inequalities, taken in conjunction with the inequalities defining the tensor products, define what we call the fusion basis. Given this basis, the machinery of our previous paper may be applied to construct the fusion generating function. New generating functions for sp(4) and su(4), together with a closed form expression for their threshold levels are presented.