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 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 $\{\infty \}$. 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 ${\mathrm{Det}}^{\prime }{\Delta }_F$ 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 ${\mathrm{Det}}^{\prime }{\Delta }_F$.

typos. arXiv admin note: text overlap with arXiv:1612.08660

References in corpus (3)

• Lowest Landau level on a cone and zeta determinants
• Spectral analysis and zeta determinant on the deformed spheres
• Metrics of constant positive curvature with conical singularities, Hurwitz spaces, and $\det \Delta$

Cited by in corpus (5)

• Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities
• Determinant of Friederichs Dirichlet Laplacians on $2$-dimensional hyperbolic cones
• Spectral determinant on Euclidean isosceles triangle envelopes of fixed area as a function of angles: absolute minimum and small-angle asymptotics
• Determinants of Laplacians for constant curvature metrics with three conical singularities on 2-sphere
• Polyakov formulas for conical singularities in two dimensions