Analytic continuation of dimensions in supersymmetric localization

Author: Anastasios Gorantis, Joseph A. Minahan and Usman Naseer

Preprint Number: UUITP-41/17

We compute the one-loop determinants for vector and hypermultiplets on spheres with d≤5 for supersymmetric gauge theories with 8 supersymmetries, showing that they are consistent with a recent conjectured form. This construction is valid for non-integer d. We then apply similar methods to gauge theories with 4 supersymmetries on spheres with d≤3, and find the one-loop determinants for vector and chiral multiplets, again valid for noninteger d. We then propose an analytic continuation from d = 3 to d = 4 that gives the perturbative partition function for an N = 1 gauge theory. We find that the results are consistent for free multiplets and with the one-loop β-functions for general N = 1 gauge theories. We also consider the analytic continuation of a mass deformation of the maximally supersymmetric gauge theory that preserves four supersymmetries and compare to recent supergravity results of Bobev et. al. for N = 1* super Yang-Mills. We find that to the extent where we can compare results, they are consistent at strong coupling, at least for the real part of the free energy.