Instantons on flat space: Explicit constructions

TL;DR

This paper revisits classic instanton solutions on flat spaces, extending known constructions from four dimensions to higher-dimensional flat manifolds with special holonomy, and explores quaternionic instantons.

Contribution

It provides explicit constructions of instantons on flat spaces of dimensions 4, 7, and 8, including less known quaternionic instantons on hypercomplex spaces.

Findings

Explicit instanton solutions on $\

Generalizations to $\

Details of quaternionic ASD instantons on $\

Abstract

In this note, we revisit some well-known examples of instantons on flat space that were originally discovered in the physics literature. In particular, we explain how the basic instanton on $\mathbb{R}^4$, with its flat hyperkaehler structure, has natural generalisations to $\mathbb{R}^7$ and $\mathbb{R}^8$ viewed as flat $G_2$- and Spin(7)-manifolds, respectively. We also provide the details of an arguably less known construction of ASD instantons on $\mathbb{H}^n$, in the sense of quaternionic geometry.