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

Abstract

We give a geometric description of the fusion rules of the affine Lie algebra s u ̂ ( 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 s u ̂ ( 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 ) × 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.