Vacuum energy on orbifold factors of spheres

Abstract

The vacuum energy is calculated for a free, conformally-coupled scalar field on the orbifold space-time \R$\times §^2/Γ$ where $Γ$ is a finite subgroup of O(3) acting with fixed points. The energy vanishes when $Γ$ 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 $ζ$- functions are analysed in some detail and several identities are incidentally derived. The general discussion is presented in terms of finite reflection groups.