  We apply the technique of localization for vertex algebras to the
Segal-Sugawara construction of an ``internal'' action of the Virasoro algebra
on affine Kac-Moody algebras. The result is a lifting of twisted differential
operators from the moduli of curves to the moduli of curves with bundles, with
arbitrary decorations and complex twistings. This construction gives a uniform
approach to a collection of phenomena describing the geometry of the moduli
spaces of bundles over varying curves: the KZB equations and heat kernels on
non-abelian theta functions, their critical level limit giving the quadratic
parts of the Beilinson-Drinfeld quantization of the Hitchin system, and their
infinite level limit giving a Hamiltonian description of the isomonodromy
equations.
