Octonionic Yang-Mills Instanton on Quaternionic Line Bundle of Spin(7) Holonomy

TL;DR

This paper constructs and classifies octonionic Yang-Mills instantons on a Spin(7) holonomy manifold derived from the quaternionic line bundle over S^4, extending self-dual instanton concepts to higher dimensions.

Contribution

It proposes an ansatz for Spin(7) Yang-Mills fields, derives associated differential equations, and classifies solutions, including explicit solutions for reduced gauge groups.

Findings

Complete solutions for specific gauge group reductions.

Classification of solutions based on asymptotic behavior.

Evidence for more general solutions via asymptotic expansion.

Abstract

The total space of the spinor bundle on the four dimensional sphere S^4 is a quaternionic line bundle that admits a metric of Spin(7) holonomy. We consider octonionic Yang-Mills instanton on this eight dimensional gravitational instanton. This is a higher dimensional generalization of (anti-)self-dual instanton on the Eguchi-Hanson space. We propose an ansatz for Spin(7) Yang-Mills field and derive a system of non-linear ordinary differential equations. The solutions are classified according to the asymptotic behavior at infinity. We give a complete solution, when the gauge group is reduced to a product of SU(2) subalgebras in Spin(7). The existence of more general Spin(7) valued solutions can be seen by making an asymptotic expansion.