Seven-dimensional super Yang-Mills at negative coupling

TL;DR

This paper explores the partition function of seven-dimensional super Yang-Mills theory on a squashed sphere, revealing instanton membrane solutions and a novel phase with tensionless membranes at negative coupling.

Contribution

It demonstrates the existence of a well-defined partition function at negative coupling and identifies a new gravitational phase with tensionless membranes.

Findings

Instanton membranes wrap fixed three-spheres on S^7.

Singularities cancel out in the full partition function.

Evidence for a tensionless membrane phase at negative coupling.

Abstract

We consider the partition function for Euclidean $SU(N)$ super Yang-Mills on a squashed seven-sphere. We show that the localization locus of the partition function has instanton membrane solutions wrapping the six "fixed" three-spheres on the $\mathbb{S}^7$. The ADHM variables of these instantons are fields living on the membrane world volume. We compute their contribution by localizing the resulting three-dimensional supersymmetric field theory. In the round-sphere limit the individual instanton contributions are singular, but the singularities cancel when adding the contributions of all six three-spheres. The full partition function on the ${\mathbb S}^7$ is well-defined even when the square of the effective Yang-Mills coupling is negative. We show for an $SU(2)$ gauge theory in this regime that the bare negative tension of the instanton membranes is canceled off by contributions from the instanton partition function, indicating the existence of tensionless membranes. We provide evidence that this phase is distinct from the usual weakly coupled super Yang-Mills and, in fact, is gravitational.