Analytic continuation of dimensions in supersymmetric localization

Abstract

We compute the perturbative partition functions for gauge theories with eight supersymmetries on spheres of dimension $d\le 5$, proving a conjecture by the second author. We apply similar methods to gauge theories with four supersymmetries on spheres with $d\le 3$. 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 $\beta$-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 $\mathcal{N}=1^{\ast }$ 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.

arXiv comment52 pages, 4 appendices, no figures, typos fixed, matches with the published version

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Cited by in corpus (14)

• $\mathcal{N}=4$ Super-Yang-Mills Correlators at Strong Coupling from String Theory and Localization
• Spherical Branes
• Chern-Simons theory from M5-branes and calibrated M2-branes
• Supersymmetric Yang-Mills, Spherical Branes, and Precision Holography
• Phases of five-dimensional supersymmetric gauge theories
• Linear Quivers at Large-$N$
• Deconstructing Little Strings with $\mathcal{N}=1$ Gauge Theories on Ellipsoids
• Perturbative solutions of $\mathcal{N}=1^{\ast }$ holography on $S^4$
• Wilson Loops and Spherical Branes
• Five-dimensional gauge theories on spheres with negative couplings
• Partition Functions of $\mathcal{N}=1$ Gauge Theories on $S^2\times {\mathbb{R}}_{\epsilon }^2$ and Duality
• (1,0) gauge theories on the six-sphere
• Gauge theories on spheres with 16 supercharges and non-constant couplings
• Tidal Love numbers and the dynamical instability of AdS bubbles