Localizing gauge theories on Sd

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 N = 2 gauge theories and d = 6 for the maximally supersymmetric case.