Casimir energy of confining large $N$ gauge theories

Four-dimensional asymptotically-free large $N$ gauge theories compactified on $S_R^3\times \mathbb{R}$ have a weakly-coupled confining regime when $R$ is small compared to the strong scale. We compute the vacuum energy of a variety of confining large $N$ non-supersymmetric gauge theories in this calculable regime, where the vacuum energy can be thought of as the $S^3$ Casimir energy. The $N=\mathrm{\infty }$ renormalized vacuum energy turns out to vanish in all of the large $N$ gauge theories we have examined, confirming a striking prediction of temperature-reflection symmetry.