Seven-dimensional super Yang-Mills at negative coupling

Authors: Joseph A. Minahan, Usman Naseer, and Charles Thull Preprint number: UUITP-32/22 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 the low energy phase in this regime is distinct from the usual weakly coupled super Yang-Mills and, in fact, is gravitational.