Theta term in a bounded region

TL;DR

This paper investigates the effects of a topological theta term in gauge theories within bounded regions, revealing non-periodic Casimir effects unless boundary terms are included, which restore periodicity and suggest minimal observable impact.

Contribution

It demonstrates how boundary terms are necessary to restore the expected periodicity of the theta term's effects in bounded gauge theories.

Findings

Casimir effect is not periodic in theta without boundary terms

Including boundary terms restores periodicity in theta

Observable effects of theta could be very small

Abstract

We analyse the physical implications of adding a topological density term $\theta Tr(F\wedge F)$ to a gauge theory in a bounded region. In particular, we calculate the Casimir effect on a spherical region and we show that the result is not periodic in $\theta$, contrary to what would be expected for a true topological density. The topological nature of the $\theta$-term can be restored if an additional boundary term required by the Atiyah-Patodi-Singer theorem is included. Then, the periodicity is trivially restored because the resulting Casimir energy is independent of $\theta$. The results of the present work suggest that the observable effects of the $\theta$-term could be very small even without assuming $\theta$ itself to be small.