(1,0) gauge theories on the six-sphere

TL;DR

This paper constructs and analyzes supersymmetric gauge theories on the six-sphere, revealing localization to Hermitian-Yang-Mills solutions and configurations constrained by complex structures, advancing understanding of higher-dimensional supersymmetric theories.

Contribution

It provides off-shell formulations of (1,0) supersymmetric gauge theories on the six-sphere and demonstrates their localization to specific geometric configurations.

Findings

Localization to Hermitian-Yang-Mills equations

Off-shell formulations for vector-multiplet and hypermultiplet

Dependence of gauge coupling on polar angle

Abstract

We construct gauge theories with a vector-multiplet and hypermultiplets of $(1,0)$ supersymmetry on the six-sphere. The gauge coupling on the sphere depends on the polar angle. This has a natural explanation in terms of the tensor branch of $(1,0)$ theories on the six-sphere. For the vector-multiplet we give an off-shell formulation for all supersymmetries. For hypermultiplets we give an off-shell formulation for one supersymmetry. We show that the path integral for the vector-multiplet localizes to solutions of the Hermitian-Yang-Mills equation, which is a generalization of the (anti-)self duality condition to higher dimensions. For the hypermultiplet, the path integral localizes to configurations where the field strengths of two complex scalars are related by an almost complex structure.