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 PYM(D) = 1 2 ∫ M ‖πHR‖θ∧(dθ), where D is a connection in a Hermitian CR-holomorphic vector bundle (E, h) → M . Let Ω = {φ < 0} ⊂ C be a smoothly bounded strictly pseuodoconvex domain and g the Bergman metric on Ω. We show that boundary valuesDb of Yang-Mills fieldsD on (Ω, g) are pseudo Yang-Mills fields on ∂Ω, provided that iTR Db = 0 and iNR D = 0 on H(∂Ω). If S → C(M) π → M is the canonical circle bundle and π∗D is a Yang-Mills field with respect to the Fefferman metric Fθ of (M, θ) then D is a pseudo Yang-Mills field on M . The Yang-Mills equations δ ∗DRπ ∗D = 0 project on the Euler-Lagrange equations δ b R D = 0 of the variational principle δ PYM(D) = 0, provided that iTR D = 0. When M has vanishing pseudohermitian Ricci curvature the pullback π∗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 {d2 PYM(D)/dt}t=0 = ∫ M 〈SD b (φ), φ〉 θ∧(dθ)n , D = D+A (provided that D is a pseudo Yang-Mills field and φ ≡ {dA/dt}t=0 ∈ Ker(δ)), and show that SD b (φ) ≡ ∆Db φ+Rb (φ), φ ∈ Ω(Ad(E)), is a subelliptic operator.