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

Abstract

We find the solution of the $\widehat{sl}(3{)}_k$ 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 $\widehat{sl}(3{)}_k$ fusion rules determined there both for generic $κ\not \in \IQ$ and fractional levels. We have also analysed the fusion in $\widehat{sl}(3{)}_k$ 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 $\widehat{sl}(2{)}_k$ fractional level admissible representations.

28 pages, 1 figure

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