Mathematical Research Letters

Volume 29 (2022)

Number 4

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

Pages: 1141 – 1163



Alexey Kokotov (Department of Mathematics and Statistics, Concordia University, Montreal, Quebec, Canada)


Let $X$ be a compact Riemann surface of genus $g\geq 2$ equipped with flat conical metric ${\lvert \Omega \rvert}$, where $\Omega$ is 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 $\Delta^{(2)}_{{\lvert \Omega \rvert}}$ below) and, respectively, anti-holomorphic line bundles ($\bar { K}^{-1}$ below). Consider the Laplace operator$\Delta^{(2)}_{{\lvert \Omega \rvert}} := {\lvert \Omega \rvert} \partial {\lvert \Omega \rvert}^{-2} \bar{\partial}$ acting in the Hilbert space of square integrable sections of the bundle $K^2$ equipped with inner product $\lt Q_1, Q_2 \gt {}_{K^2} = \int_X\frac {Q_1\bar Q_2}{{\lvert \Omega \rvert}}$.

We discuss two natural definitions of the determinant of the operator $\Delta^{(2)}_{{\lvert \Omega \rvert}}$. 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. Considering the regularized determinant of $\Delta^{(2)}_{{\lvert \Omega \rvert}}$ 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 $g$.

Received 18 January 2020

Accepted 2 June 2020

Published 23 February 2023