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 6manifolds. 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.

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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.