The Compatibility between the Higher Dimensions Self Duality and the Yang-Mills Equation of Motion

TL;DR

This paper investigates the relationship between higher-dimensional self-duality conditions and the Yang-Mills equations, revealing that self-duality implies the equations only in four dimensions, but remains approximately compatible in higher dimensions for small instantons.

Contribution

It demonstrates the compatibility of higher-dimensional self-duality with Yang-Mills equations, extending the understanding beyond four dimensions and highlighting the mathematical utility of self-duality.

Findings

Self-duality implies Yang-Mills equations only in 4D for generic instanton sizes.

In higher dimensions, self-duality is approximately compatible for small instantons.

Self-duality simplifies second order equations to first order, aiding mathematical analysis.

Abstract

We study the compatiblity between the higher dimension dualities and the Yang-Mills equation of motion. Taking a 't Hooft solution as a starting point, we come to the conclusion that for only 4 dimensions the self duality implies the equation of motion for generic instanton size. Whereas in higher dimensions, the self duality is compatable with the equation of motion, approximately, for small instanton size i.e. the zero curvature condition. At the mathematical level, the self duality is still useful since it transforms a second order into a first order differential equation.