Yang–Mills theory

Remarks

In Riemannian (Euclidean) signature one uses instead $S_E(A) = \frac{1}{2 g_{\mathrm{YM}}^2} \int_M \abs{F_A}^2 \, \mathrm{vol}_g$, the Yang–Mills functional. Sign conventions for the Lorentzian action differ between books; the field equations do not depend on them. A Yang–Mills theory with matter adds terms for fields in associated bundles, as in quantum chromodynamics.

Example: $G = U(1)$ with $\mathfrak{g} = i\R$. Then $F_A = dA$ is a closed form, $d_A = d$ on forms with values in $\mathfrak{g}$, and the Yang–Mills equations become $d \star F = 0$, Maxwell's equations in vacuum. For a non-abelian $G$ the curvature in a local trivialization is $F = dA + \frac{1}{2}[A \wedge A]$, and the quadratic term makes the equations nonlinear.

Non-example: on a $3$-manifold, take the Chern–Simons action, which up to a constant factor is $\int_M \ip{A}{\wedge dA} + \frac{1}{3} \ip{A}{\wedge [A \wedge A]}$. It is built from $A$ itself and not from $\abs{F_A}^2$, so it does not define a Yang–Mills theory. Its field equations say that $A$ is flat.

In physics notation, with $\mathfrak{g}$ realized by Hermitian matrices and $\mathrm{tr}(T^a T^b) = \frac{1}{2}\delta^{ab}$, the Lorentzian Lagrangian density is $-\frac{1}{2}\mathrm{tr}(F_{\mu\nu} F^{\mu\nu}) = -\frac{1}{4} F^a_{\mu\nu} F^{a\,\mu\nu}$ after rescaling $A$ by the coupling.