Theta term in a bounded region

We analyse the physical implications of adding a topological density term $θTr(F\wedge F)$ 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 $θ$, contrary to what would be expected for a true topological density. The topological nature of the $θ$-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 $θ$. The results of the present work suggest that the observable effects of the $θ$-term could be very small even without assuming $θ$ itself to be small.

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