Self-dual Yang-Mills equations ($F=\ast F$)

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.

In oriented Euclidean four-space, gauge curvature is self-dual when $F=\ast F$. The Hodge star squares to one on two-forms, so this picks its positive eigenspace; the negative eigenspace is anti-self-duality of gauge curvature. The Bianchi identity then implies the Yang-Mills equations, and the action saturates the positive-charge Yang-Mills instanton Bogomolny bound.

With orientation $d{x}^1\wedge d{x}^2\wedge d{x}^3\wedge dt$, ${ϵ}_{123}=1$, and temporal gauge $A_t=0$, the mixed gauge curvature is $F_{ti}={\partial }_t{A}_i$. The self-dual Yang-Mills equations then have the displayed form. Placing $dt$ first in the orientation reverses the sign, so the metric alone does not specify it.