Twisted determinants on higher genus Riemann surfaces

We study the Dirac and the Laplacian operators on orientable Riemann surfaces of arbitrary genus g. In particular we compute their determinants with twisted boundary conditions along the b-cycles. All the ingredients of the final results (including the normalizations) are explicitly written in terms of the Schottky parametrization of the Riemann surface. By using the bosonization equivalence, we derive a multi-loop generalization of the well-known g=1 product formulae for the Theta-functions. We finally comment on the applications of these results to the perturbative theory of open charged strings.