Five-dimensional gauge theories on spheres with negative couplings

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.

arXiv comment34 pages, 11 figures

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• Wilson Loops and Spherical Branes
• Seven-dimensional super Yang-Mills at negative coupling