Vacuum energy on orbifold factors of spheres

TL;DR

This paper calculates the vacuum energy for a scalar field on orbifolded spheres, revealing it vanishes for pure rotation groups and providing general formulas involving Bernoulli functions and reflection groups.

Contribution

It provides a general analysis of vacuum energy on orbifold spheres, including explicit formulas and identities for the associated zeta functions, extending previous results to even-dimensional cases.

Findings

Vacuum energy vanishes for pure rotation orbifolds.

Vacuum energies expressed as Todd polynomials (generalized Bernoulli functions).

Detailed analysis of zeta functions related to orbifold symmetries.

Abstract

The vacuum energy is calculated for a free, conformally-coupled scalar field on the orbifold space-time \R$\times \S^2/\Gamma$ where $\Gamma$ is a finite subgroup of O(3) acting with fixed points. The energy vanishes when $\Gamma$ is composed of pure rotations but not otherwise. It is shown on general grounds that the same conclusion holds for all even-dimensional factored spheres and the vacuum energies are given as generalised Bernoulli functions (i.e. Todd polynomials). The relevant $\zeta$- functions are analysed in some detail and several identities are incidentally derived. The general discussion is presented in terms of finite reflection groups.