Effective action in spherical domains

The effective action on an orbifolded sphere is computed for minimally coupled scalar fields. The results are presented in terms of derivatives of Barnes ζ–functions and it is shown how these may be evaluated. Numerical values are shown. An analytical, heat-kernel derivation of the Cesàro-Fedorov formula for the number of symmetry planes of a regular solid is also presented. 1 1. Introduction. In earlier work [1] we have shown that the ζ–function, ζ Γ (s), on orbifold-factored spheres, S d /Γ, for a conformally coupled scalar field, is given by a Barnes ζ–function, [2], ζ d (s, a | d), where the d i are the degrees associated with the tiling group Γ. The free-field Casimir energy on the space-time R×S d /Γ was given as the value of the ζ– function at a negativ e integer which evaluated to a generalised Bernoulli function. In the present work we wish to consider the effective action on orbifolds S d /Γ which this time are to be looked upon as Euclidean space-times. In particular we will discuss d = 2 and d = 3, concentrating on the former. The simplifying assumption in our previous work was that of conformal coupling on R×S d /Γ. This made the relevant eigenvalues perfect squares and allowed us to use known generating functions to incorporate the degeneracies. From the point of view of field theories on the space-times S d /Γ, retaining this assumption would be rather artificial. A more appropriate choice would be minimal coupling, or possibly conformal coupling, on S d /Γ. (These coincide for d = 2.) The quantities in which we are interested are ζ