Functional determinants on regions of the plane and sphere

TL;DR

This paper extends the conformal transformation formula for effective action changes to piecewise smooth boundaries and computes functional determinants of the scalar Laplacian on specific planar and spherical regions, including sectors and caps.

Contribution

It introduces a generalized formula for effective action changes with piecewise smooth boundaries and calculates determinants on regions derived from orbifolded spheres.

Findings

Derived a generalized conformal transformation formula for piecewise smooth boundaries.

Computed functional determinants for sectors and crescent-shaped regions.

Determined the effective action on spherical caps.

Abstract

The standard formula for the change in the effective action under a conformal transformation is extended to the case when the boundary is piecewise smooth. We then find the functional determinants of the scalar Laplacian on regions of the plane obtained by stereographic projection of the fundamental domains on an orbifolded two-sphere. Examples treated are the sector of a disk and a circular crescent. The effective action on a spherical cap is also determined.