For a family of smooth curves, we have the associated family
of moduli spaces of stable bundles with fixed determinant on the
curves. There exists a so-called theta line bundle on the family of
moduli spaces. When the Kodaira-Spencer map of the family of
curves is an isomorphism, we prove in this paper an identification
theorem between sheaves of differential operators on the theta line
bundle and higher direct images of vector bundles on curves. As an
application, the so-called Hitchin connection on the direct image
of (powers of) the theta line bundle is derived naturally from the
identification theorem. A logarithmic extension to certain singular
stable curves is also presented in this paper.