A Polyakov formula for sectors

TL;DR

This paper derives a variational formula for how the zeta-regularized determinant of the Laplacian on convex Euclidean sectors changes with the sector's opening angle, involving explicit heat kernel and determinant computations.

Contribution

It introduces a Polyakov formula for sectors, explicitly computes heat kernels, and analyzes the extremal properties of determinants on rectangular domains.

Findings

Explicit formula for the heat kernel on infinite sectors

Variation of the Laplacian determinant with sector angle

Square maximizes the determinant among rectangles

Abstract

We consider finite area convex Euclidean circular sectors. We prove a variational Polyakov formula which shows how the zeta-regularized determinant of the Laplacian varies with respect to the opening angle. Varying the angle corresponds to a conformal deformation in the direction of a conformal factor with a logarithmic singularity at the origin. We compute explicitly all the contributions to this formula coming from the different parts of the sector. In the process, we obtain an explicit expression for the heat kernel on an infinite area sector using Carslaw-Sommerfeld's heat kernel. We also compute the zeta-regularized determinant of rectangular domains of unit area and prove that it is uniquely maximized by the square.