Self-dual solutions to pseudo Yang–Mills equations

Abstract

We study pseudo Yang–Mills fields on a compact 5-dimensional strictly pseudoconvex CR manifold M i.e. critical points to the functional Y Mb(D)=12∫M‖ΠHRD‖2θ∧(dθ)2 on the space C(E,h) of all connections D on a Hermitian vector bundle (E,h) over M, such that Dh=0. If A={D∈C(E,h):ξ⌋RD=0,Gθ∗(Tr(RD),dθ)=0} and D∈A is an absolute minimum to Y Mb:A→R then (i) ΔbTr(RD)=0 and (ii) D is self-dual or anti-self-dual according to the sign of c2(θ,D)=∫Mθ∧{P2(D)−m−12mP1(D)∧P1(D)} [where Pk(D) is the k-th Chern form of (E,D)] and provided c2(θ,D) is constant on A.