Title: Flat conical Laplacian in the square of the canonical bundle and its regularized determinants
Abstract: We discuss two natural definitions of the determinant of the Dolbeault Laplacian acting in the square of the canonical bundle over a compact Riemann surface equipped with flat conical metric given by the modulus of a holomorphic quadratic differential with simple zeroes. The first one uses the zeta-function of some special self-adjoint extension of the Laplacian (initially defined on smooth sections vanishing near the zeroes of the quadratic differential), the second one is an analog of Eskin-Kontsevich-Zorich (EKZ) regularization of the determinant of the conical Laplacian acting in the trivial bundle. In contrast to the situation of operators acting in the trivial bundle, these two regularizations turn out to be essentially different. Considering the regularized determinant of the Laplacian as a functional on the moduli space of quadratic differentials with simple zeroes on compact Riemann surfaces of a given genus, we derive explicit expressions for this functional for the both regularizations. The expression for the EKZ regularization is closely related to the well-known explicit expressions for the Mumford measure on the moduli space of compact Riemann surfaces.