Yang–Mills functional

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

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

Definition (Yang–Mills functional)

The Yang–Mills functional is

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.

Critical points of this functional are precisely Yang–Mills connections , characterized by the Yang–Mills equation .

Examples

1. Flat connections. If $F_A=0$ then $\mathrm{YM}(A)=0$, which is the minimum possible value.
2. 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 $\frac12\int |F|^2$.
3. Four dimensions and self-duality. On an 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).