Flat surfaces and determinants of Laplacians

In 2008 D. Korotkin and the author found an explicit formula for the determinant of the Friedrichs Laplacian on a compact 2d surface of genus g >1 provided with flat conical metric with trivial holonomy. This formula can be considered as a higher genus version of the classical Ray-Singer formula for the determinant of the Laplace operator on an elliptic curve  with (smooth) flat conformal metric.
We will discuss further generalizations of this result for
1) General polyhedral metrics on compact surfaces.
2) Other (i.e. non Friedrichs) self-adjoint extensions of conical Laplacians.
3) Noncompact flat surfaces with cylindrical ends (Mandelstam diagrams)
4)Noncompact flat surfaces  with Euclidean or conical ends (Hurwitz diagrams).

The latter case turns out to be closely related to the theory of Hurwitz Frobenius manifolds, in particular, the isomonodromic tau-function of the Hurwitz Frobenius manifold makes its striking appearance in the explicit formulas for the determinant. 

The talk is based on the joint works with L. Hillairet and V. Kalvin.