Affine su(2) fusion rules from gerbe 2-isomorphisms

TL;DR

This paper provides a geometric framework using gerbes to describe the fusion rules of the affine Lie algebra su(2)_k, linking algebraic fusion coefficients to gerbe 2-isomorphisms over conjugacy classes.

Contribution

It introduces a novel geometric approach to derive su(2)_k fusion rules through gerbe 2-isomorphisms, connecting algebraic and geometric perspectives.

Findings

Fusion rules expressed via gerbe 2-isomorphisms.

Gerbe trivializations correspond to dominant weights.

Application to junctions of defect lines in WZW models.

Abstract

We give a geometric description of the fusion rules of the affine Lie algebra su(2)_k at a positive integer level k in terms of the k-th power of the basic gerbe over the Lie group SU(2). The gerbe can be trivialised over conjugacy classes corresponding to dominant weights of su(2)_k via a 1-isomorphism. The fusion-rule coefficients are related to the existence of a 2-isomorphism between pullbacks of these 1-isomorphisms to a submanifold of SU(2) x SU(2) determined by the corresponding three conjugacy classes. This construction is motivated by its application in the description of junctions of maximally symmetric defect lines in the Wess-Zumino-Witten model.