Five-dimensional gauge theories on spheres with negative couplings

TL;DR

This paper explores supersymmetric gauge theories on five-spheres with negative couplings, revealing novel phase structures and effective theories, and analyzing their free energy and instanton contributions in the large N limit.

Contribution

It provides a detailed analysis of gauge theories with negative couplings on $S^5$, including their localization, phase behavior, and effective Chern-Simons descriptions, extending understanding of non-standard coupling regimes.

Findings

Pure ${ m SU}(N)$ theory reduces to a product of Chern-Simons theories at negative coupling.

Eigenvalue density width in ${ m SU}(N/2)_{N/2}$ Chern-Simons theory is exactly 1.

One-instanton contributions remain exponentially suppressed at negative coupling in large N limit.

Abstract

We consider supersymmetric gauge theories on $S^5$ with a negative Yang-Mills coupling in their large $N$ limits. Using localization we compute the partition functions and show that the pure ${\mathrm{SU}}(N)$ gauge theory descends to an ${\mathrm{SU}}(N/2)_{+N/2}\times {\mathrm{SU}}(N/2)_{-N/2}\times {\mathrm{SU}}(2)$ Chern-Simons gauge theory as the inverse 't Hooft coupling is taken to negative infinity for $N$ even. The Yang-Mills coupling of the ${\mathrm{SU}}(N/2)_{\pm N/2}$ is positive and infinite, while that on the ${\mathrm{SU}}(2)$ goes to zero. We also show that the odd $N$ case has somewhat different behavior. We then study the ${\mathrm{SU}}(N/2)_{N/2}$ pure Chern-Simons theory. While the eigenvalue density is only found numerically, we show that its width equals $1$ in units of the inverse sphere radius, which allows us to find the leading correction to the free energy when turning on the Yang-Mills term. We then consider ${\mathrm{USp}}(2N)$ theories with an antisymmetric hypermultiplet and $N_f<8$ fundamental hypermultiplets and carry out a similar analysis. Along the way we show that the one-instanton contribution to the partition function remains exponentially suppressed at negative coupling for the ${\mathrm{SU}}(N)$ theories in the large $N$ limit.