Yang–Mills Theory
Definition. Gauge theory with nonabelian structure group $G$ ($SU(2)$, $SU(3)$, …): a connection on a principal $G$-bundle with action $$S = -\frac{1}{2g^2}\int \mathrm{tr}\,(F \wedge \star F), \qquad F = dA + \tfrac12[A, A].$$ Equations of motion: $d_\nabla \star F = \star J$; Bianchi: $d_\nabla F = 0$ (covariant $d$'s now, not plain $d$).
Electromagnetism is the abelian special case. Set $G = U(1)$: $[A,A]=0$, trace trivial ⇒ Landau's $-\frac{1}{16\pi c}\int F_{ik}F^{ik}\,d\Omega$ (§27). The entire structure of Landau's Ch. 3–4 survives; what changes for nonabelian $G$:
| $U(1)$ (Maxwell/Landau) | Nonabelian (Yang–Mills) | |
|---|---|---|
| $F$ | $dA$ — linear | $dA + \frac12[A,A]$ — nonlinear |
| Gauge field self-coupling | none (photons neutral) | yes (gluons colored — carry adjoint charge) |
| $F$ under gauge transf. | invariant | covariant only: $g^{-1}Fg$ |
| Superposition | holds | fails — the theory is intrinsically nonlinear |
Physics anchor. The Standard Model = connections on a $U(1) \times SU(2) \times SU(3)$ bundle + matter in specified representations. Everything in this vault is the $U(1)$ row of that table.
Related: Curvature 2-Form · Adjoint Representation · SU(2) and SO(3) · Characteristic Classes (instantons live here) · Module 10 - Representations and Matter Fields
Sources: Landau §27–§30 (abelian case); Woit (final chapters).