Fusion in coset CFT from admissible singular-vector decoupling

TL;DR

This paper addresses the challenge of accurately deriving coset fusion rules from WZW models at fractional levels, proposing methods to reconcile discrepancies between different computational approaches.

Contribution

It introduces two prescriptions that produce correct coset fusion rules from decoupling methods, specifically for Virasoro minimal models and the su(2) case.

Findings

Corrected coset fusion rules from decoupling methods

Reconciliation of Verlinde formula and decoupling results

Application to Virasoro minimal models

Abstract

Fusion rules for Wess-Zumino-Witten (WZW) models at fractional level can be defined in two ways, with distinct results. The Verlinde formula yields fusion coefficients that can be negative. These signs cancel in coset fusion rules, however. On the other hand, the fusion coefficients calculated from decoupling of singular vectors are non-negative. They produce incorrect coset fusion rules, however, when factorisation is assumed. Here we give two prescriptions that yield the correct coset fusion rules from those found for the WZW models by the decoupling method. We restrict to the Virasoro minimal models for simplicity, and because decoupling results are only complete in the $\su(2)$ case.