First order deconfinement transition in largeNYang-Mills theory on a smallS3

We give an analytic demonstration that the $3+1$-dimensional large $N$ $SU(N)$ pure Yang-Mills theory, compactified on a small $S^3$ so that the coupling constant at the compactification scale is very small, has a first order deconfinement transition as a function of temperature. We do this by explicitly computing the relevant terms in the canonical partition function up to three-loop order; this is necessary because the leading (one-loop) result for the phase transition is precisely on the border line between a first order and a second order transition. Since numerical work strongly suggests that the infinite-volume large $N$ theory also has a first order deconfinement transition, we conjecture that the phase structure is independent of the size of the $S^3$. To deal with divergences in our calculations, we are led to introduce a novel method of regularization useful for non-Abelian gauge theory on $S^3$.