Some asymptotics of Topological quantum field theory via skein theory

Abstract

For each oriented surface $\Sigma$ of genus $g$, we study a limit of quantum representations of the mapping class group arising in topological quantum field theory (TQFT) derived from the Kauffman bracket. We determine that these representations converge in the Fell topology to the representation of the mapping class group on $H(\Sigma )$, the space of regular functions on the $\mathrm{SL}(2,\mathbb{C})$-representation variety with its Hermitian structure coming from the symplectic structure of the $\mathrm{SU}(2)$-representation variety. As a corollary, we give a new proof of the asymptotic faithfulness of quantum representations