Invariant Yang-Mills connections over Non-Reductive Pseudo-Riemannian Homogeneous Spaces

TL;DR

This paper classifies invariant gauge fields over certain non-reductive pseudo-Riemannian homogeneous spaces and identifies conditions under which these fields are Yang-Mills, revealing a unique universal invariant connection in an exceptional case.

Contribution

It provides a classification of G-invariant principal bundles and connections over non-reductive homogeneous spaces, introducing a new notion of reductivity and analyzing the principle of symmetric criticality.

Findings

All G-invariant connections on classified bundles are Yang-Mills in two special cases.

PSC fails in most cases due to degeneracy of scalar product on symmetric variations.

A unique universal G-invariant connection exists in one exceptional case.

Abstract

We study invariant gauge fields over the 4-dimensional non-reductive pseudo-Riemannian homogeneous spaces G/K recently classified by Fels & Renner (2006). Given H compact semi-simple, classification results are obtained for principal H-bundles over G/K admitting: (1) a G-action (by bundle automorphisms) projecting to left multiplication on the base, and (2) at least one G-invariant connection. There are two cases which admit nontrivial examples of such bundles and all G-invariant connections on these bundles are Yang-Mills. The validity of the principle of symmetric criticality (PSC) is investigated in the context of the bundle of connections and is shown to fail for all but one of the Fels-Renner cases. This failure arises from degeneracy of the scalar product on pseudo-tensorial forms restricted to the space of symmetric variations of an invariant connection. In the exceptional case where PSC is valid, there is a unique G-invariant connection which is moreover universal, i.e. it is the solution of the Euler-Lagrange equations associated to any G-invariant Lagrangian on the bundle of connections. This solution is a canonical connection associated with a weaker notion of reductivity which we introduce.