Einstein and Yang-Mills implies conformal Yang-Mills
Samuel Blitz, A. Rod Gover, Jarosław Kopiński, Andrew Waldron
TL;DR
The paper develops a conformally invariant, higher‑derivative analogue of the Yang–Mills condition for connections on vector bundles over even‑dimensional conformal manifolds with $n\ge 6$, via a conformal Yang–Mills current $k[g,A]$ and an energy $E_{\rm CYM}[[g],A]$. Using tractor calculus and holography, it relates the boundary CYM current of the tractor connection to the Fefferman–Graham obstruction tensor $\mathcal{B}$ through a bundle map $k[c,A^{\mathcal{T}}] = 2(n-1)(n-2) q_{\sf s}(\mathcal{B})$, so that $\mathcal{B}=0$ iff the CYM current vanishes. The results show that the Einstein and Yang–Mills conditions imply the CYM condition, and establish asymptotic Yang–Mills behavior in the bulk with the obstruction tensor governing the boundary CYM data. The paper also provides explicit examples demonstrating that CYM is a strict weakening of YM and that the tractor approach yields tractable, invariant formulations of these higher‑derivative conditions.
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 Poincaré-Einstein manifold is itself Yang-Mills.