Localizing gauge theories on $S^d$

We conjecture the form of the one-loop determinants for localized gauge theories with eight supersymmetries on $d$-dimensional spheres. Combining this with results for the localized action, we investigate the strong coupling behavior in the large $N$ limit for a continuous range of $d$. In particular, we find the $N$ dependence of the free energy for supersymmetric Yang-Mills with only a vector multiplet in $3<d<4$ and for maximally supersymmetric Yang-Mills in $3<d<6$. We also argue that this gives an effective way to regularize divergences after localization in $d=4$ for $\mathcal{N}=2$ gauge theories and $d=6$ for the maximally supersymmetric case.