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 $\Bbb CP^1$. On a compact Riemann surface $X$ of genus $g$ consider a meromorphic funciton $f: X\to {\Bbb C}P^1$ 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 $\operatorname{Det}' \Delta_F$ 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 $\operatorname{Det}' \Delta_F$.