On Determinants of Laplacians on Compact Riemann Surfaces Equipped with Pullbacks of Conical Metrics by Meromorphic Functions

Abstract

Let ${\mathsf {m}}$ be any conical (or smooth) metric of finite volume on the Riemann sphere ${\mathbb {C}}P^1$. On a compact Riemann surface X of genus g consider a meromorphic function $f: X\rightarrow {{\mathbb {C}}}P^1$ such that all poles and critical points of f are simple and no critical value of f coincides with a conical singularity of ${\mathsf {m}}$ or $\{\infty \}$. The pullback $f^*{\mathsf {m}}$ of ${\mathsf {m}}$ under f has conical singularities of angles $4\pi$ at the critical points of f and other conical singularities that are the preimages of those of ${\mathsf {m}}$. We study the $\zeta$-regularized determinant ${\text {Det}}^\prime \Delta _F$ of the (Friedrichs extension of) Laplace–Beltrami operator on $(X,f^*{\mathsf {m}})$ as a functional on the moduli space of pairs (X, f) and obtain an explicit formula for ${\text {Det}}^\prime \Delta _F$.