Yang–Mills functional

Let $M$ be an oriented Riemannian manifold, let $G$ be a Lie group with an $\mathrm{Ad}$ -invariant positive-definite inner product on its Lie algebra (as for a compact structure group), and let $\pi\colon P\to M$ be a principal G-bundle.

Fix a principal connection $A$ on $P$ with curvature $F_A\in \Omega^2(M;\operatorname{ad}(P))$ .

The Yang–Mills functional is

\mathrm{YM}(A) := \frac12\int_M |F_A|^2\,\mathrm{vol}_g\in[0,+\infty],

where the norm combines the Riemannian metric with the chosen invariant inner product; the integral may be infinite on a noncompact base.

Equivalent formula and properties

Equivalently, $\mathrm{YM}(A)=\frac12\int_M \langle F_A\wedge *F_A\rangle$ using the Hodge star of the Riemannian metric and the chosen inner product.

It is invariant under gauge transformations of $P$ , so it descends to a functional on the moduli space of connections modulo gauge.

On a closed manifold, critical points of this functional are precisely Yang–Mills connections, characterized by the Yang–Mills equation.

Examples

Flat connections. If $F_A=0$ then $\mathrm{YM}(A)=0$ , which is the minimum possible value.
Abelian case (Maxwell energy). For $G=U(1)$ , the curvature is an ordinary closed 2-form, and $\mathrm{YM}(A)$ reduces to the classical electromagnetic energy $\frac12\int |F|^2$ .
Four dimensions and self-duality. On a closed oriented 4-manifold, connections with self-dual or anti-self-dual curvature minimize $\mathrm{YM}$ within their topological class (the functional splits into a topological term plus a nonnegative remainder).