Asymptotic faithfulness of the quantum SU(n) representations of the mapping class groups

Abstract

We prove that the sequence of projective quantum ${\rm SU}(n)$ representations of the mapping class group of a closed oriented surface, obtained from the projective flat ${\rm SU}(n)$-Verlinde bundles over Teichmüller space, is asymptotically faithful. That is, the intersection over all levels of the kernels of these representations is trivial, whenever the genus is at least $3$. For the genus $2$ case, this intersection is exactly the order $2$ subgroup, generated by the hyper-elliptic involution, in the case of even degree and $n=2$. Otherwise the intersection is also trivial in the genus $2$ case.