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

Abstract

We define a conformally invariant action [Formula: see text] on gauge connections on a closed pseudo-Riemannian manifold [Formula: see text] of dimension 6. At leading order this is quadratic in the gauge connection. The Euler–Lagrange equations of [Formula: see text], 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 [Formula: see text] on [Formula: see text], we define [Formula: see text] by first defining a Lagrangian density associated to [Formula: see text]. This is not conformally invariant but has a conformal transformation analogous to a [Formula: see text]-curvature. Integrating this density provides the conformally invariant action. In the special case that we apply [Formula: see text] 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.