Deformed $G_2$-instantons on $\mathbb{R}^4 \times S^3$

TL;DR

This paper constructs the first explicit non-trivial examples of deformed $G_2$-instantons, also known as Donaldson-Thomas connections, on specific $G_2$ manifolds, advancing understanding of gauge theory in higher dimensions.

Contribution

It provides the first explicit non-trivial examples of deformed $G_2$-instantons on particular $G_2$ manifolds, including $ ext{R}^4 imes S^3$ and $ ext{R}^+ imes S^3 imes S^3$.

Findings

Explicit deformed $G_2$-instantons constructed on $ ext{R}^4 imes S^3$ and $ ext{R}^+ imes S^3 imes S^3$.

Identification of an associative foliation of $ ext{R}^4 imes S^3$ by $ ext{R}^2 imes S^1$.

Abstract

We construct explicit examples of deformed $G_2$-instantons, also called Donaldson-Thomas connections, on $\mathbb{R}^4 \times S^3$ endowed with the torsion free $G_2$-structure found by Brandhuber et al. and on $\mathbb{R}^+\times S^3 \times S^3$ endowed with the Bryant-Salamon conical $G_2$-structure. These are the first such non-trivial examples on a $G_2$ manifold. As a by-product of our investigation we also find an associative foliation of $\mathbb{R}^4\times S^3$ by $\mathbb{R}^2 \times S^1$.