Variational formulas for determinant of Laplacian on higher genus polyhedral surface

TL;DR

This work derives variational formulas for the zeta-regularized determinant of the Friedrichs Laplacian on higher-genus Riemann surfaces equipped with flat polyhedral (conical) metrics, expressing how det$\Delta$ changes with conical points and angles. Starting from perturbation theory for eigenvalues and heat-kernel asymptotics near conical singularities, the authors obtain an infinitesimal Polyakov-type formula for $\log(\det\Delta)$ and integrate it to yield an explicit determinant expression up to a moduli-dependent constant. They compute the relevant cone-geometry integrals $\mathfrak{I}(\beta)$ and $\tilde{\mathfrak{I}}(\beta)$, derive the coefficients $A_j$ and $\tilde{A}_j$ through several canonical deformations (varying vertex positions, scale, cone angles), and present genus-1 specialization with explicit theta-function and Dedekind-eta-function forms. The results generalize known trivial-holonomy formulas and provide a concrete, computationally accessible framework for det$\Delta$ on polyhedral surfaces, including a torus case that recovers the Ray-Singer formula via $\mathbf{c}_0(\mathbb{B})=\Im\mathbb{B}\,|\eta(\mathbb{B})|^4$.

Abstract

Let $X$ be a Riemann surface of genus $g\ge 1$ endowed with a flat conical metric $m$ and let ${\rm det}\,Δ$ be the $ζ$-regularized determinant of the Friedrichs Laplacian on $(X,m)$. We derive variational formulas for ${\rm det}\,Δ$ with respect to conical points and conical angles within a given conformal class. Integration of them leads to an explicit expression for ${\rm det}\,Δ$ up to moduli dependent factor. The latter, in principle, can be calculated via comparison of the above result with the well-known formulas for the case of flat conical metrics with trivial holonomy.