With E. Frenkel. **Geometric Realization of the
Segal-Sugawara Construction**.

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.