Einstein and Yang-Mills implies conformal Yang-Mills

TL;DR

This paper introduces conformally invariant higher-derivative analogs of the Yang-Mills condition, linking them to Einstein metrics and the Fefferman-Graham obstruction tensor, thus extending Yang-Mills theory in conformal geometry.

Contribution

It provides a compact formula for conformal Yang-Mills analogs and establishes their relation to Einstein metrics and the Fefferman-Graham obstruction tensor.

Findings

Conformal Yang-Mills analogs are a strict weakening of the Yang-Mills condition under Einstein metrics.

The conformal Yang-Mills condition for the tractor connection is equivalent to the vanishing of the Fefferman-Graham obstruction tensor.

The tractor connection on a Poincaré-Einstein manifold is itself Yang-Mills.

Abstract

There exist conformally invariant, higher-derivative, variational analogs of the Yang-Mills condition for connections on vector bundles over a conformal manifold of even dimension greater than or equal to six. We give a compact formula for these analogs and prove that they are a strict weakening of the Yang-Mills condition with respect to an Einstein metric. We also show that the conformal Yang-Mills condition for the tractor connection of an even dimensional conformal manifold is equivalent to vanishing of its Fefferman-Graham obstruction tensor. This result uses that the tractor connection on a Poincar\'e-Einstein manifold is itself Yang-Mills.