A note on decoupling conditions for generic level $\hat{sl}(3)_k$ and fusion rules

TL;DR

This paper solves specific decoupling equations for $ ext{hat} sl(3)_k$ in 3-point functions, confirming fusion rules and clarifying previous algebraic approaches, especially for fractional levels.

Contribution

It provides explicit solutions to $ ext{hat} sl(3)_k$ decoupling equations and validates fusion rules for generic and fractional levels, extending prior algebraic methods.

Findings

Explicit solutions involve hypergeometric functions with singularities.

Confirmed fusion rules for generic and fractional levels.

Clarified previous treatments of $ ext{hat} sl(2)_k$ fusion at fractional levels.

Abstract

We find the solution of the $\hat{sl}(3)_k$ singular vector decoupling equations on 3-point functions for the particular case when one of the fields is of weight $w_0\cdot k\Lambda_0$. The result is a function with non-trivial singularities in the flag variables, namely a linear combination of 2F1 hypergeometric functions. This calculation fills in a gap in [1] and confirms the $\hat{sl}(3)_k$ fusion rules determined there both for generic $\kappa \not \in \IQ$ and fractional levels. We have also analysed the fusion in $\hat{sl}(3)_k$ using algebraic methods generalising those of Feigin and Fuchs and again find agreement with [1]. In the process we clarify some details of previous treatments of the fusion of $\hat{sl}(2)_k$ fractional level admissible representations.