On a compact complex variety, the Hirzebruch-Riemann-Roch formula relates de holomorphic Euler characteristic of a vector bundle in terms of characteristic classes. The family version of this theorem admits a differential form lifting proven by Bismut-Gillet-Soulé,, computing the curvature of the Quillen metric on the determinant of the relative cohomology in terms of charscteristic differential forms of hermitian vector bundles (Chern-Weil theory). Sometimes vector bundles come with connections (e.g. flat connections) which are not necessarily compatible with a metric. What would be the analohue of the Quillen metric and Chern-Weil characteristic forms? In this talk I will discuss what can be done for families of Riemann surfaces and line bundles endowed with flat connections. The talk will be based on joint work with Richard Wentworth.