Yang-Mills connections of cohomogeneity one on SO(n)-bundles over Euclidean spheres

TL;DR

This paper constructs explicit Yang-Mills connections on SO(n)-bundles over spheres with Euclidean metrics by reducing the problem to ODEs via symmetry, and demonstrates solutions exist for various bundles.

Contribution

It introduces a method to find Yang-Mills connections on bundles over spheres using cohomogeneity one symmetry and variational techniques, expanding known solutions.

Findings

Solutions for Yang-Mills connections on SO(6)-bundles over S^6.

Existence of Yang-Mills connections on tangent bundles of spheres for n=5 to 9.

Application of harmonic map ideas to solve the reduced ODE system.

Abstract

We construct Yang-Mills connections on SO(n)-bundles over spheres equipped with the Euclidean metric. We use a cohomogeneity one group action on the bundle to reduce the Yang-Mills-equation to a system of ordinary differential equations. The system is shown to have solutions by variational methods, using ideas from harmonic map theory. Examples include Yang-Mills connections on each of the countably many principal SO(6)-bundles over $S^6$, and countably many Yang-Mills connections on $TS^n$ for $n\in\{5,...,9\}$.