Flat metrics with conical singularities on
Riemann surfaces and determinants of Laplacians

Alexey Kokotov

Abstract: Flat metrics with conical singularities on compact Riemann surfaces are considered.
The metrics are given as the 2/n-th power of the modulus of holomorphic (meromorphic) n-differentials.
Our goal is to calculate the zeta-regularized determinants of the Laplacians corresponding to these metrics.
The determinant of the Laplacian essentially coincides (up to a simple factor) with the modulus squared of a
holomorphic function on the moduli space of n-differentials. This function turns out to be a complete analog
of the isomonodromic tau-function of Hurwitz Frobenius manifolds.