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.