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

The aim of this paper is to determine the existence condition of the moduli space of $SU(3)$-flat connections on $3$-holed $2$-sphere $D$, the so-called pair of pants, and to study its relationship to the $\widehat {\mathfrak{sl}}(3;\mathbb{C})$ 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.