A conformally invariant Yang-Mills type energy and equation on 6-manifolds

TL;DR

This paper introduces a new conformally invariant energy functional for gauge connections on 6-manifolds, leading to higher-order equations that generalize Yang-Mills theory and relate to conformal curvature invariants.

Contribution

It defines a novel conformally invariant action in six dimensions and links its Euler-Lagrange equations to the Fefferman-Graham obstruction tensor, extending known 4D results.

Findings

The action is conformally invariant and quadratic in the gauge connection.

Euler-Lagrange equations recover the obstruction-flat condition for 6-manifolds.

Special case with the Cartan-tractor connection relates to the Fefferman-Graham obstruction tensor.

Abstract

We define a conformally invariant action S on gauge connections on a closed pseudo-Riemannian manifold M of dimension 6. At leading order this is quadratic in the gauge connection. The Euler-Lagrange equations of S, with respect to variation of the gauge connection, provide a higher-order conformally invariant analogue of the (source-free) Yang-Mills equations. For any gauge connection A on M, we define S(A) by first defining a Lagrangian density associated to A. This is not conformally invariant but has a conformal transformation analogous to a Q-curvature. Integrating this density provides the conformally invariant action. In the special case that we apply S to the conformal Cartan-tractor connection, the functional gradient recovers the natural conformal curvature invariant called the Fefferman-Graham obstruction tensor. So in this case the Euler-Lagrange equations are exactly the "obstruction-flat" condition for 6-manifolds. This extends known results for 4-dimensional pseudo-Riemannian manifolds where the Bach tensor is recovered in the Yang-Mills equations of the Cartan-tractor connection.