The moduli space of 𝑆𝑈(3)-flat connections and the fusion rules

Abstract

The aim of this paper is to determine the existence condition of the moduli space of S U ( 3 ) SU(3) -flat connections on 3 3 -holed 2 2 -sphere D D , the so-called pair of pants, and to study its relationship to the s l ^ ( 3 ; C ) \widehat {\mathfrak {sl}}(3;\mathbb {C}) fusion rules. The existence condition can be expressed by a system of inequalities with the entries of highest weights with respect to the fundamental weights. This gives a necessary condition for the fusion coefficents to be nontrivial. We also find that the fusion coefficient of a triplet of extremal highest weights equals one. This can be considered a quantum counterpart of the PRV-conjecture.