Hitchin's connection and differential operators with values in the determinant bundle

Abstract

Let $C/M$ be a local universal family of smooth curves and $S/M$ be the family of moduli spaces of stable bundles with a fixed determinant on curves. In this paper, we find locally free sheaves $\Cal G_E$, $S(\Cal G_E)$ on $X=C\times_M S$ such that their first direct images are isomorphic to sheaves $\Cal D^{\le 1}_{S/M}(\Theta)$, $\Cal D^{\le 1}_S(\Theta)$ of 1-st order differential operators on the theta line bundle over $S$. As an application, we give a new construction of Hitchin's projective connection (or KZ-connection). Our main results have clearly an extension to some stable singular curves. Then we construct a logarithmic projective connection (in fact, a logarithmic projective heat operator on the theta line bundle) that extends Hitchin's connection to a coherent sheaf over an open set (with at least codimension two) of the moduli space of stable curves. Such an extension seems not reachable by other methods (as far as we know).