 Self-dual Yang-Mills equations

The self-dual Yang-Mills equations require a gauge curvature two-form to have positive Hodge star operator eigenvalue on an oriented Riemannian four-manifold. By the Bianchi identity, they imply the full second-order Yang-Mills equations. Smooth finite-action solutions saturate the Yang-Mills instanton Bogomolny bound. Reversing orientation exchanges this equation with the Anti-self-dual Yang-Mills equations.