  We use the theory of Berezin-Toeplitz operators of Ma and Marinescu to study
the spaces of holomorphic sections of a prequantizing line bundle over compact
K\"ahler manifolds under deformations of the complex structure. We show that
the parallel transport in the induced vector bundle over the deformation space
behaves like a Toeplitz operator, and compute its first coefficient. We then
use this result to establish a semi-classical trace formula for the induced
quantization of symplectic maps, and give an application to Witten's asymptotic
expansion conjecture for the quantum representations of the mapping class
group.
