SU(2) solutions to self-duality equations in eight dimensions

TL;DR

This paper constructs explicit solutions to octonionic self-duality equations in eight dimensions on specific manifolds, reducing to non-abelian Seiberg-Witten equations in four dimensions, revealing singularities and regular solutions in particular geometric backgrounds.

Contribution

It provides explicit solutions to octonionic self-duality equations on eight-dimensional manifolds and their reductions to non-abelian Seiberg-Witten equations, including regular solutions on nilpotent Lie groups.

Findings

Solutions are singular on flat and Eguchi-Hanson backgrounds.

A regular solution exists on a nilpotent Lie group with a conformally hyper-K"ahler metric.

Explicit construction of solutions on manifolds of the form M_4×R^4.

Abstract

We consider the octonionic self-duality equations on eight-dimensional manifolds of the form $M_8=M_4\times \R^4$, where $M_4$ is a hyper-K\"ahler four-manifold. We construct explicit solutions to these equations and their symmetry reductions to the non-abelian Seiberg-Witten equations on $M_4$ in the case when the gauge group is SU(2). These solutions are singular for flat and Eguchi-Hanson backgrounds. For $M_4=\R\times {\mathcal G}$ with a cohomogeneity one hyper-K\"ahler metric, where ${\mathcal G}$ is a nilpotent (Bianchi II) Lie group, we find a solution which is singular only on a single-sided domain wall. This gives rise to a regular solution of the non-abelian Seiberg-Witten equations on a four-dimensional nilpotent Lie group which carries a regular conformally hyper-K\"ahler metric.