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 $F_A$.

Theorem/Definition (Yang–Mills equation)

The Euler–Lagrange equation for the Yang–Mills functional is

Here $d_A$ is the covariant exterior derivative on $\mathrm{Ad}(P)$-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.

Examples

1. Flat connections. If $F_A=0$ then $d_A(*F_A)=0$ automatically.
2. Abelian reduction. For $G=U(1)$, the equation becomes $d(*F)=0$, the source-free Maxwell equation for the curvature 2-form $F$.
3. 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 $d_A(*F_A)=\pm d_AF_A=0$.