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

Abstract

Let $f:X\to \mathbb{C}{P}^1$ 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 $\mathbb{C}{P}^1$. Then the pullback $f^{\ast }\mathsf{m}$ 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 $f^{\ast }\mathsf{m}$ 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.

References in corpus (2)

• Lowest Landau level on a cone and zeta determinants
• Spectral analysis and zeta determinant on the deformed spheres

Cited by in corpus (3)

• Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities
• On Determinants of Laplacians on Compact Riemann Surfaces Equipped with Pullbacks of Conical Metrics by Meromorphic Functions
• Determinants of Laplacians for constant curvature metrics with three conical singularities on 2-sphere