On an infinitesimal Polyakov formula for genus zero polyhedra

2025/04/24 by Alexey Kokotov, Dmitrii Korikov, Kokotov, Alexey +1 · 1 citation
Engineering · Mathematics · #30F10 #30F45 #32G08 #35P99 #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Primary 58J52 #Secondary 32G15 #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2504.17652

openalex publication_date 2025/04/24 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

Abstract

Let X be a genus zero compact polyhedral surface (the Riemann sphere equipped with a flat conical metric m). We derive the variational formulas for the determinant of the Laplacian, \rm det Δ^m, on X under infinitesimal variations of the positions of the conical points and the conical angles (i. e. infinitesimal variations of X in the class of polyhedra with the same number of vertices). Besides having an independent interest, this derivation may serve as a somewhat belated mathematical counterpart of the well-known heuristic calculation of \rm det Δ^m performed by Aurell and Salomonson in the 90-s.

Citations

• On a polygon version of Wiegmann-Zabrodin formula
• Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities
• Lowest Landau level on a cone and zeta determinants
• A heat trace anomaly on polygons
• Holomorphic factorization of determinants of laplacians on Riemann surfaces and a higher genus generalization of Kronecker's first limit formula
• Tau-functions on spaces of Abelian differentials and higher genus generalizations of Ray-Singer formula
• Further results on Functional Determinants of Laplacians in Simplicial Complexes
• Effective action in spherical domains
• On Functional Determinants of Laplacians in Polygons and Simplices
• Extremals of determinants of Laplacians

Cited by

• Variational formulas for determinant of Laplacian on higher genus polyhedral surface

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• The determinant on flat conic surfaces with excision of disks