Flat conical Laplacian in the square of the canonical bundle and its regularized determinants

Abstract

Let $X$ be a compact Riemann surface of genus $g\geq 2$ equipped with flat conical metric $|\Omega|$, where $\Omega$ be a holomorphic quadratic differential on $X$ with $4g-4$ simple zeroes. Let $K$ be the canonical line bundle on $X$. Introduce the Cauchy-Riemann operators $\bar \partial$ and $\partial$ acting on sections of holomorphic line bundles over $X$ ($K^2$ in the definition of $\Delta^{(2)}_{|\Omega|}$ below) and, respectively, anti-holomorphic line bundles ($\bar { K}^{-1}$ below). Consider the Laplace operator $\Delta^{(2)}_{|\Omega|}:=|\Omega| \partial |\Omega|^{-2}\bar\partial$ acting in the Hilbert space of square integrable sections of the bundle $K^2$ equipped with inner product $ _{K^2}=\int_X\frac {Q_1\bar Q_2}{|\Omega|}$. We discuss two natural definitions of the determinant of the operator $\Delta^{(2)}_{|\Omega|}$. The first one uses the zeta-function of some special self-adjoint extension of the operator (initially defined on smooth sections of $K^2$ vanishing near the zeroes of $\Omega$), 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, for operators acting in $K^2$ these two regularizations turn out to be essentially different. Considering the regularized determinant of $\Delta^{(2)}_{|\Omega|}$ as a functional on the moduli space $Q_g(1, \dots, 1)$ of quadratic differentials with simple zeroes on compact Riemann surfaces of genus $g$, 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 of genus $g$.