Moduli spaces of $G_{2}$ and $Spin(7)-$instantons on product manifolds

TL;DR

This paper establishes a correspondence between $G_2$ and $Spin(7)$-instantons on product manifolds and Hermitian Yang-Mills connections on the base manifold, revealing their topological moduli structure.

Contribution

It demonstrates that $G_2$-instantons on $X imes S^1$ correspond to Hermitian Yang-Mills connections on $X$, and extends similar results to $Spin(7)$-instantons in dimension 8.

Findings

$G_2$-instantons are equivalent to Hermitian Yang-Mills connections on $X$

The moduli space of $G_2$-instantons is topologically characterized

Extension of results to $Spin(7)$-instantons in dimension 8

Abstract

Let $X$ be a closed $6-$dimensional manifold with a half-closed $SU(3)-$structure. On the product manifold $X\times S^{1}$, with respect to the product $G_{2}-$structure and on a pullback vector bundle from $X$, we show that any $G_{2}-$instanton is equivalent to a Hermitian Yang-Mills connection on $X$ via a "broken gauge". This result reveals the topological type of the moduli of $G_{2}-$instantons on $X\times S^{1}$. In dimension $8$, similar result holds for moduli of $Spin(7)-$instantons. A generalization and an example are given.