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

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 funciton $f:X\to \mathbb{C}{P}^1$ 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 $\{\mathrm{\infty }\}$. The pullback $f^{\ast }\mathsf{m}$ 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 ${\mathrm{Det}}^{\mathrm{\prime }}{\mathrm{\Delta }}_F$ of the (Friedrichs extension of) Laplace-Beltrami operator on $(X,f^{\ast }\mathsf{m})$ as a functional on the moduli space of pairs $(X,f)$ and obtain an explicit formula for $\operatorname{Det}’ \Delta_F$.