Analytic continuation of dimensions in supersymmetric localization

We compute the perturbative partition functions for gauge theories with eight supersymmetries on spheres of dimension d <= 5, proving a conjecture by the second author. We apply similar methods to gauge theories with four supersymmetries on spheres with d <= 3. 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 N = 1 gauge theory. The results are consistent with the free multiplets and the one-loop beta-functions for general N = 1 gauge theories. We also consider the analytic continuation of an N = 1 preserving mass deformation of the maximally supersymmetric gauge theory and compare to recent holographic results for 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