Analytic continuation of dimensions in supersymmetric localization

We compute the perturbative partition functions for gauge theories with eight supersymmetries on spheres of dimension $d\le5$, proving a conjecture by the second author. We apply similar methods to gauge theories with four supersymmetries on spheres with $d\le3$. The results are valid for non-integer $d$ as well. We further propose an analytic continuation from $d=3$ to $d=4$ that gives the perturbative partition function for an $\mathcal{N}=1$ gauge theory. The results are consistent with the free multiplets and the one-loop $\beta$-functions for general $\mathcal{N}=1$ gauge theories. We also consider the analytic continuation of an $\mathcal{N}=1$-preserving mass deformation of the maximally supersymmetric gauge theory and compare to recent holographic results for $\mathcal{N}=1^*$ super Yang-Mills. We find that the general structure for the real part of the free energy coming from the analytic continuation is consistent with the holographic results.