Casimir Domains on a sphere

TL;DR

This paper analyzes Casimir effects on a sphere divided by latitude lines for scalars and spinors, revealing symmetry-driven minima and domain behaviors, with explicit formulas and connections to previous work.

Contribution

It introduces a Casimir analysis for spherical domains divided by latitude lines, including explicit effective actions and symmetry considerations for scalars and spinors.

Findings

Effective action minima occur at symmetrical arrangements.

Dirichlet slice expands to fill the sphere.

Neumann slice stabilizes at approximately 58 degrees.

Abstract

A Casimir--type analysis of the effect of dividing the two--sphere by several lines of latitude is done for conformally invariant Dirichlet and Neumann scalars and for spinors. An effective action combination is shown to have minima for symmetrical arrangements of the circles. For a domain of two caps and a slice, the Dirichlet slice expands to fill the whole sphere while the Neumann one adjusts to an angular separation of 57.92 degrees. The fermion expression is written in terms of the Weber class function, $\gf$, and a connection is noted with an earlier calculation of twisted scalar effective actions on the tetrahedron.