Yang–Mills Equations
in Differential‑Form Language

Core objects

• A – 𝔤‑valued 1‑form (gauge potential)
• F – 𝔤‑valued 2‑form (curvature)
• D – covariant exterior derivative D = d + [A, ·]
• * – Hodge star (metric‑dependent)

Definitions

The wedge product carries the Lie‑algebra commutator, i.e. A∧A = ½ [A_μ , A_ν] dx^μ∧dx^ν.

Field equation

‑ Vacuum (J = 0): D *F = 0.
‑ In 4‑D Minkowski space, *F is also a 2‑form, giving the familiar component form D_μ F^{μν}=0.

Compact summary

F   = dA + A∧A
DF  = 0
D*F = *J