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

Abstract

We consider the octonionic self-duality equations on eight-dimensional manifolds of the form $M_8=M_4\times {\mathbb{R}}^4$, where $M_4$ is a hyper-Kähler 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 $M_4=\mathbb{R}\times \mathcal{G}$ with a cohomogeneity one hyper-Kähler 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-Kähler metric.

Dedicated to Jerzy Lukierski on the occasion of his 75th birthday. Final version, to appear in JGP

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