#
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.