  In this paper, we will provide a review of the geometric construction,
proposed by Witten, of the SU(n) quantum representations of the mapping class
groups which are part of the Reshetikhin-Turaev TQFT for the quantum group
U_q(sl(n, C)). In particular, we recall the differential geometric construction
of Hitchin's projectively flat connection in the bundle over Teichmuller space
obtained by push-forward of the determinant line bundle over the moduli space
of rank n, fixed determinant, semi-stable bundles fibering over Teichmuller
space. We recall the relation between the Hitchin connection and Toeplitz
operators which was first used by the first named author to prove the
asymptotic faithfulness of the SU(n) quantum representations of the mapping
class groups. We further review the construction of the formal Hitchin
connection, and we discuss its relation to the full asymptotic expansion of the
curve operators of Topological Quantum Field Theory. We then go on to
identifying the first terms in the formal parallel transport of the Hitchin
connection explicitly. This allows us to identify the first terms in the
resulting star product on functions on the moduli space. This is seen to agree
with the first term in the star product on holonomy functions on these moduli
spaces defined by Andersen, Mattes and Reshetikhin.
