Moduli spaces of meromorphic functions and determinant of Laplacian

TL;DR

This paper investigates the determinant of the Laplace operator on moduli spaces of meromorphic functions on Riemann surfaces, providing explicit formulas and analyzing its properties through PDEs, special functions, and spectral theory.

Contribution

It introduces a regularized determinant for the Laplace operator on Hurwitz spaces and derives explicit formulas relating it to classical objects on Riemann surfaces, along with new decomposition and variational formulas.

Findings

Explicit expression for the Laplacian determinant in terms of prime form and theta-functions.

Decomposition formula for Laplacian determinants on flat surfaces with conical singularities.

Variational formulas for eigenvalues as conical singularities move.

Abstract

The Hurwitz space is the moduli space of pairs $(X,f)$ where $X$ is a compact Riemann surface and $f$ is a meromorphic function on $X$. We study the Laplace operator $\Delta^{|df|^2}$ of the flat singular Riemannian manifold $(X,|df|^2)$. We define a regularized determinant for $\Delta^{|df|^2}$ and study it as a functional on the Hurwitz space. We prove that this functional is related to a system of PDE which admits explicit integration. This leads to an explicit expression for the determinant of the Laplace operator in terms of the basic objects on the underlying Riemann surface (the prime form, theta-functions, the canonical meromorphic bidifferential) and the divisor of the meromorphic differential $df$. The proof has several parts that can be of independent interest. As an important intermediate result we prove a decomposition formula of the type of Burghelea-Friedlander-Kappeler for the determinant of the Laplace operator on flat surfaces with conical singularities and Euclidean or conical ends. We introduce and study the $S$-matrix, $S(\lambda)$, of a surface with conical singularities as a function of the spectral parameter $\lambda$ and relate its behavior at $\lambda=0$ with the Schiffer projective connection on the Riemann surface $X$. We also prove variational formulas for eigenvalues of the Laplace operator of a compact surface with conical singularities when the latter move.