Yang–Mills equation

Let $P\to M$ be a principal $G$ -bundle over an oriented Riemannian manifold, and let $A$ be a principal connection with curvature $F_A$ .

The Yang–Mills equation is

d_A(*F_A)=0.

Here $*$ is the Hodge star, and $d_A$ is the covariant exterior derivative on $\operatorname{ad}(P)$ -valued forms, which extends the exterior derivative on ordinary forms and satisfies the Bianchi identity $d_A F_A=0$ .

A connection $A$ satisfying $d_A(*F_A)=0$ is called a Yang–Mills connection.

Variational interpretation

If the Lie algebra carries an Ad-invariant positive-definite inner product, this is the Euler–Lagrange equation of the Yang–Mills functional on a closed manifold. On a noncompact manifold the same local equation follows for finite-energy connections under compactly supported variations; boundaries require boundary conditions or variations supported away from the boundary.

Examples

Flat connections. If $F_A=0$ then $d_A(*F_A)=0$ automatically.
Abelian reduction. For $G=U(1)$ , the equation becomes $d(*F)=0$ , the source-free Maxwell equation for the curvature 2-form $F$ .
Instantons in dimension 4. On a 4-manifold, any connection with self-dual or anti-self-dual curvature satisfies the Yang–Mills equation because $*F_A=\pm F_A$ and the Bianchi identity gives $d_A(*F_A)=\pm d_AF_A=0$ .