  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.
