Instantons on the exceptional holonomy manifolds of Bryant and Salamon

TL;DR

This paper constructs $G_2$ and $Spin(7)$ instantons on Bryant and Salamon's exceptional holonomy manifolds using a spherical symmetry ansatz, revealing asymptotic behavior related to Hermitian Yang-Mills connections.

Contribution

It introduces a new method for constructing instantons on these manifolds by exploiting their bundle structure and symmetry properties.

Findings

Constructed explicit $G_2$ and $Spin(7)$ instantons on Bryant-Salamon manifolds.

Showed asymptotic convergence of $G_2$ instantons to Hermitian Yang-Mills connections.

Provided insights into the geometric structure of instantons on exceptional holonomy spaces.

Abstract

We give a construction of $G_2$ and $Spin(7)$ instantons on exceptional holonomy manifolds constructed by Bryant and Salamon, by using an ansatz of spherical symmetry coming from the manifolds being the total spaces of rank-4 vector bundles. In the $G_2$ case, we show that, in the asymptotically conical model, the connections are asymptotic to Hermitian Yang-Mills connections on the nearly K\"ahler $S^3\times S^3$.