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$\times \S^2/\Gamma$ 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.Comment: 29 pages, late

Elliptic aspects of statistical mechanics on spheres

“…[2,3], yield more specific forms. We begin here with an expression given by Chang and Dowker [4] which is somewhat more general than just for the sphere. The situation there discussed was a conformal scalar on an orbifold factoring ('triangulation') of the sphere, written S d /Γ, where Γ is a regular solid symmetry group.…”

Spinors and forms on the ball and the generalized cone

“…The idea is that this gives the 'square-root' heatkernel, and the sphere ^-function, (14), follows by Mellin transform on r as in our earlier works, [18,13]. The generating function for a given form order can be rewritten using the identity (41)…”

“…The first sum reduces to a term that is cancelled by the zero mode and to a single Barnes C-fimction, the result being, [13], [8] …”

p -form spectra and Casimir energies on spherical tessellations

“…To illustrate the techniques I look first at the scalar field and repeat some material from [8], which uses degrees, and then outline the angle sum approach which can be applied to the spin-1 case. Later, I will describe the degree approach for the Maxwell field after dealing with p-forms.…”

“…I would rather consider the hemisphere as a (simple) example of a fundamental domain of a regular spherical tesselation and continue with the calculations begun in [8]. This reference deals with scalar fields, but Chang's thesis, [9], contains higher spin results and I will be incorporating these since the works [1,3] are concerned specifically with Maxwell fields.…”