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.