The Prym-Hitchin connection 

Christian Pauly
Université de Nice
In the first part of this talk I will outline the construction of Hitchin's projective connection on the sheaf of non-abelian level-
theta functions over the moduli space of semi-stable rank-
vector bundles over a family of smooth complex projective curves via an algebro-geometric approach using heat operators developped by Welters, Hitchin, van Geemen- de Jong. Then, I will briefly explain why the monodromy representation of the Hitchin connection has infinite image, except in a small number of cases. This result stands in contrast with the finiteness of the monodromy of abelian theta functions. I will concentrate on one of these exceptional cases, namely
. In that case, Prym varieties, i.e. anti-invariant loci of Jacobians of curves equipped with an involution, and their non-abelian analogues, studied by Zelaci, naturally appear in the monodromy problem. Finally, I will present recent results extending the construction of Hitchin's connection to families of curves with involutions. This is joint work with Baier, Bolognesi and Martens.