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

Hitchin's Projectively Flat Connection, Toeplitz Operators and the Asymptotic Expansion of TQFT Curve Operators