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 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 (Xf) and obtain an explicit formula for \({\text {Det}}^\prime \Delta _F\).


Conical metric Determinants of Laplacians Moduli space 

Mathematics Subject Classification




It is a pleasure to thank Alexey Kokotov for helpful discussions and important remarks.


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© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsConcordia UniversityMontrealCanada

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