Self-dual solutions to pseudo Yang–Mills equations

We study pseudo Yang–Mills fields on a compact 5-dimensional strictly pseudo convex CR manifold $M$ i.e. critical points to the functional $\mathcal{Y M}_b(D) = 1/2 \int_M \| \Pi_H R^D \|^2 \theta \wedge (d\theta)^2$ on the space $\mathcal{C}(E, h)$ of all connections $D$ on a Hermitian vector bundle $(E, h)$ over $M$, such that $Dh = 0$. If $\mathcal{A} = \{D \in \mathcal{C}(E, h) : \xi \lfloor R^D = 0$, $G^∗_\theta (Tr(R^D), d\theta) = 0\}$ and $D \in \mathcal{A}$ is an absolute minimum to $\mathcal{Y M}_b : \mathcal{A} \to \mathbb{R}$ then (i) $\Delta_b Tr(R^D) = 0$ and (ii) $D$ is self-dual or anti-self-dual according to the sign of $c_2 (\theta, D) = \int_M \theta \wedge{\mathbf{P}_2 (D) − \frac{m−1}{2m} \mathbf{P}_1(D) \wedge \mathbf{P}_1(D)\}$ [where $\mathbf{P}_k(D)$ is the $k$-th Chern form of $(E, D)$] and provided $c_2(\theta, D)$ is constant on $\mathcal{A}$.