p-form spectra and Casimir energies on spherical tesselations

TL;DR

This paper computes Casimir energies for scalar and Maxwell fields on spherical tesselations of the three-sphere, develops spectral theory for p-forms, and explores heat-kernel expansions with topological insights.

Contribution

It introduces a spectral theory for p-forms on spherical tesselations and analyzes Casimir energies with boundary conditions and heat-kernel properties.

Findings

Casimir energies computed for scalar and Maxwell fields.

Degeneracy generating functions for p-forms derived.

Heat-kernel expansion terminates with a constant term of 1/2.

Abstract

Casimir energies on space-times having the fundamental domains of semi-regular spherical tesselations of the three-sphere as their spatial sections are computed for scalar and Maxwell fields. The spectral theory of p-forms on the fundamental domains is also developed and degeneracy generating functions computed. Absolute and relative boundary conditions are encountered naturally. Some aspects of the heat-kernel expansion are explored. The expansion is shown to terminate with the constant term which is computed to be 1/2 on all tesselations for a coexact 1-form and shown to be so by topological arguments. Some practical points concerning generalised Bernoulli numbers are given.