Abstract
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 mathcal{N} =1 gauge theory. The results are consistent with the free multiplets and the one-loop β-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.
Highlights
Localization has proven to be a powerful tool for investigating supersymmetric gauge theories on compact spaces with isometries
We further 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 general structure for the real part of the free energy coming from the analytic continuation is consistent with the holographic results
Summary
Localization has proven to be a powerful tool for investigating supersymmetric gauge theories on compact spaces with isometries (for a recent review see [1]). In [10] a conjecture was given for the partition function of supersymmetric gauge theories in the zero instanton sector on round spheres with eight supersymmetries, for general dimension d. The three dimensional mass deformed gauge theory that we can analytically continue requires real masses Such terms appear explicitly as central charges in the superalgebra. The presence of the cubic term in the superpotential forces the sum of the three real masses to be zero in order to maintain supersymmetry Despite these subtleties, one can compare the general structure of the analytically continued partition function with the N = 1∗ partition function.
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