On twisted N=2 5D super Yang-Mills theory

TL;DR

This paper constructs a twisted N=2 5D super Yang-Mills theory on Sasaki-Einstein and K-contact manifolds, relating contact instantons to elliptic equations and providing a covariant framework for the Haydys-Witten equations.

Contribution

It introduces a new twisted 5D super Yang-Mills theory on Sasaki-Einstein manifolds, connecting contact instantons with elliptic equations and extending the understanding of supersymmetric gauge theories.

Findings

Localization on contact instantons via elliptic equations

Reduction of Haydys-Witten equations to contact instantons under certain conditions

Construction of a covariant version of Haydys-Witten equations

Abstract

On a five dimensional simply connected Sasaki-Einstein manifold, one can construct Yang-Mills theories coupled to matter with at least two supersymmetries. The partition function of these theories localises on the contact instantons, however the contact instanton equations are not elliptic. It turns out that these equations can be embedded into the Haydys-Witten equations (which are elliptic) in the same way the 4D anti-self-dual instanton equations are embedded in the Vafa-Witten equations. We show that under some favourable circumstances, the latter equations will reduce to the former by proving some vanishing theorems. It was also known that the Haydys-Witten equations on product manifolds $M_5=M_4\times \mathbb{R}$ arise in the context of twisting the 5D maximally supersymmetric Yang-Mills theory. In this paper, we present the construction of twisted $N=2$ Yang-Mills theory on Sasaki-Einstein manifolds, and more generally on $K$-contact manifolds. The localisation locus of this new theory thus provides a covariant version of the Haydys-Witten equation.