  We prove a formula for the determinant of Laplacian on an arbitrary compact
polyhedral surface of genus one. This formula generalizes the well-known
Ray-Singer result for a flat torus. A special case of flat conical metrics
given by the modulus of a meromorphic quadratic differential on an elliptic
surface is also considered. We study the determinant of Laplacian as a
functional on the moduli space of meromorphic quadratic differentials with L
simple poles and L simple zeros and derive formulas for variations of this
functional w. r. t. natural coordinates on this moduli space. A new proof of
the Troyanov theorem about flat conical metrics on compact Riemann surfaces of
arbitrary genus is also given.
