  In this paper we provide a review of asymptotic results of Toeplitz operators
and their applications in TQFT. To do this we review the differential geometric
construction of the Hitchin connection on a prequantizable compact symplectic
manifold. We use asymptotic results relating the Hitchin connec- tion and
Toeplitz operators, to, in the special case of the moduli space of flat
SU(n)-connections on a surface, prove asymptotic faithfulness of the SU(n)
quantum representations of the mapping class group. We then go on to re- view
formal Hitchin connections and formal trivializations of these. We discuss how
these fit together to produce a Berezin-Toeplitz star product, which is
independent of the complex structure. Finally we give explicit examples of all
the above objects in the case of the abelian moduli space. We furthermore
discuss an approach to curve operators in the TQFT associated to abelian
Chern-Simons theory.
