Yang-Mills fields on CR manifolds

We study pseudo Yang–Mills fields on a compact strictly pseudoconvex CR manifold $M$, i.e., the critical points of the functional $\mathcal{P Y M}(D)= \frac{1}{2} \int_M ||\pi_H R^D ||^2 \theta \wedge (d \theta )^n$ where $D$ is a connection in a Hermitian CR holomorphic vector bundle $(E,h) \to M$. Let $\Omega = \{ \varphi < 0\} \subset \mathbb{C}^n$ be a smoothly bounded strictly pseuodoconvex domain and $g$ the Bergman metric on $\Omega$. We show that boundary values $D_b$ of Yang-Mills fields $D$ on $(\Omega , g)$ are pseudo Yang-Mills fields on $\partial \Omega$, provided that $i_T R^{D_b}=0$ and $i_N R^D = 0$ on $H(\partial \Omega)$. If $S^1 \to C(M) \to M$ is the canonical circle bundle and $\pi^* D$ is a Yang-Mills field with respect to the Fefferman metric $F_\theta$ of $(M, \theta)$, then $D$ is a pseudo Yang-Mills field on $M$. The Yang-Mills equations $\delta^{\pi^* D} R^{\pi^* D} = 0$ project on the Euler-Lagrange equations $\delta_b^D R^D = 0$ of the variational principle $ \delta \mathcal{P Y M} (D) = 0$, provided that $i_T R^D =0$. When $M$ has vanishing pseudohermitian Ricci curvature the pullback $\pi^* D$ of the (CR invariant) Tanaka connection $D$ of $(E,h)$ is a Yang-Mills field on $C(M)$. We derive the second variation formula $\{d^2 \mathcal{P Y M} (D^t)/ d t^2\}_{t=0} = \int_M \langle S_b^D (\varphi) , \varphi \rangle \theta \wedge (d \theta )^n$, $D^t = D + A^t$ [provided that $D$ is a pseudo Yang-Mills field and $\varphi \equiv \{ d A^t/dt\}_{t=0} \in \mathrm{Ker} (\delta^D)$], and show that $S_b^D (\varphi) \equiv \Delta_b^D \varphi + \mathcal{R}_b^D (\varphi )$ $\varphi \in \Omega^{0,1} [\mathrm{Ad}(E)]$, is a subelliptic operator.