Metrics of constant positive curvature with conical singularities, Hurwitz spaces, and ${\rm det}\, \Delta$

TL;DR

This paper derives an explicit formula for the zeta-regularized determinant of the Laplace operator on Riemann surfaces with metrics of constant positive curvature and conical singularities, parametrized by Hurwitz spaces.

Contribution

It introduces a novel explicit formula for the Laplace operator determinant on surfaces with conical singularities arising from meromorphic functions, linking geometry and spectral theory.

Findings

Explicit formula for the Laplace determinant on Hurwitz spaces

Connection between conical singularities and spectral invariants

Advancement in understanding metrics with conical singularities

Abstract

Let $f: X\to {\Bbb C}P^1$ be a meromorphic function of degree $N$ with simple poles and simple critical points on a compact Riemann surface $X$ of genus $g$ and let $\mathsf m$ be the standard round metric of curvature $1$ on the Riemann sphere ${\Bbb C}P^1$. Then the pullback $f^*\mathsf m$ of $\mathsf m$ under $f$ is a metric of curvature $1$ with conical singularities of conical angles $4\pi$ at the critical points of $f$. We study the $\zeta$-regularized determinant of the Laplace operator on $X$ corresponding to the metric $f^*\mathsf m$ as a functional on the moduli space of the pairs $(X, f)$ (i.e. on the Hurwitz space $H_{g, N}(1, \dots, 1)$) and derive an explicit formula for the functional.