Analytic continuation of dimensions in supersymmetric localization

TL;DR

This paper develops a method to analytically continue the partition functions of supersymmetric gauge theories across different dimensions, providing new insights and consistency checks with holographic results.

Contribution

It introduces an analytic continuation approach for supersymmetric gauge theories' partition functions across non-integer dimensions, extending previous conjectures and connecting to holographic computations.

Findings

Partition functions computed for theories with 8 and 4 supersymmetries on spheres of various dimensions.

Proposed an analytic continuation from 3D to 4D for  gauge theories.

Results align with known free multiplets,  beta-functions, and holographic predictions.

Abstract

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.