Yang–Mills connection

Let $P\to M$ be a principal $G$ -bundle over an oriented Riemannian manifold.

A principal connection $A$ on $P$ is called a Yang–Mills connection if it satisfies the Yang–Mills equation

d_A(*F_A)=0,

where $F_A$ is its curvature.

Variational interpretation

If the Lie algebra carries an Ad-invariant positive-definite inner product, the equation is equivalent to being a critical point of the Yang–Mills functional on a closed base. For a noncompact base, assume finite energy and use compactly supported variations; with boundary, impose boundary conditions or keep variations away from the boundary.

Examples

Flat connections. Any flat connection is Yang–Mills, since its curvature vanishes.
Anti-self-dual connections. On an oriented 4-manifold, ASD (or SD) connections are Yang–Mills; these are the basic instanton solutions in gauge theory.
Constant curvature on surfaces. On a closed oriented surface, the Yang–Mills equation says that the curvature coefficient $*F_A$ is covariantly constant. For an irreducible connection this coefficient is central; for reducible connections it may lie in a larger holonomy centralizer.