the witten-reshetikhin-turaev topological quantum field theory in particular provides us with the so-called quantum representations of mapping class groups. the geometric construction of these involves geometric quantization of moduli spaces, which produced a holomorphic vector bundle over teichm\"uller space. this bundle supports a projectively flat connection constructed by algebraic geometric techniques by hitchin. we will present a a toeplitz operator approximation formula for the parallel transport of the hitchin connection. we will discuss applications of this