Presentations of Wess-Zumino-Witten Fusion Rings

TL;DR

This paper characterizes the fusion rings of Wess-Zumino-Witten models, distinguishing between fusion rings over integers and algebras over complex numbers, and provides explicit presentations for various models.

Contribution

It offers complete proofs for the fusion algebras of SU(r+1) and Sp(2r), constructs generators for general WZW fusion rings over Z, and extends the formalism to spin groups.

Findings

Fusion algebras over C are characterized by fusion potentials for SU(r+1) and Sp(2r).

Explicit generators for WZW fusion rings over Z are constructed.

Fusion rings for spin groups of odd rank are explicitly presented.

Abstract

The fusion rings of Wess-Zumino-Witten models are re-examined. Attention is drawn to the difference between fusion rings over Z (which are often of greater importance in applications) and fusion algebras over C. Complete proofs are given characterising the fusion algebras (over C) of the SU(r+1) and Sp(2r) models in terms of the fusion potentials, and it is shown that the analagous potentials cannot describe the fusion algebras of the other models. This explains why no other representation-theoretic fusion potentials have been found. Instead, explicit generators are then constructed for general WZW fusion rings (over Z). The Jacobi-Trudy identity and its Sp(2r) analogue are used to derive the known fusion potentials. This formalism is then extended to the WZW models over the spin groups of odd rank, and explicit presentations of the corresponding fusion rings are given. The analogues of the Jacobi-Trudy identity for the spinor representations (for all ranks) are derived for this purpose, and may be of independent interest.