Tau-functions on spaces of holomorphic differentials over Riemann surfaces and determinants of Laplacians in flat metrics with conic singularities over Riemann surfaces

The main goal of this paper is to compute (up to a moduli-independent constant factor) determinants of Laplacians in flat metrics with conic singularities on compact Riemann surfaces. We consider two classes of metrics: the Ströbel metrics and metrics given by moduli square of a holomorphic differential. For the latter case, if all conic angles equal 4π, our formulas essentially coincide with heuristic expressions proposed by Sonoda in 1987. For this goal we develop the formalism of Bergmann tau-functions on moduli spaces of holomorphic one and twodifferentials on Riemann surfaces; these tau-functions are natural analogs of isomonodromic tau-functions from the theory of Hurwitz Frobenius manifolds. As a by-product we get a new version of Rauch variational formulas for the spaces of holomorphic differentials. In our computations we essentially rely on techniques developed by Fay in his 1992 memoir.