---
tag: B6K478R9XE
kind: definition
name: Yang–Mills functional
msc: ["58E15", 53C07, 81T13]
defines: CKGDR9P9PA
uses: [7CBT4SM83Y, 7JBRGCVJRB, 8E37E74XVP, NJ21PN5ZGJ, ZAJFW1S078, JJWETW5AT8]
sources:
  - url: https://en.wikipedia.org/wiki/Yang–Mills_equations
---

## Definition

Let $(M, g)$ be a compact oriented [[Riemannian manifold|7CBT4SM83Y]] without boundary, with volume form $\mathrm{vol}_g$.
Let $G$ be a compact [[Lie group|7JBRGCVJRB]] with Lie algebra $\mathfrak{g}$.
Fix an inner product on $\mathfrak{g}$ that is invariant under the adjoint action of $G$.
Let $P \to M$ be a [[principal $G$-bundle|8E37E74XVP]], and let $\operatorname{ad} P = P \times_G \mathfrak{g}$ be its adjoint bundle.
For a [[connection|NJ21PN5ZGJ]] $A$ on $P$, let $F_A$ be its [[curvature|ZAJFW1S078]], a $2$-form on $M$ with values in $\operatorname{ad} P$.
The metric $g$ and the inner product on $\mathfrak{g}$ give a pointwise norm $\abs{F_A}$.
The *Yang–Mills functional* is the map from the space of connections on $P$ to $[0, \infty)$ given by
$$\mathrm{YM}(A) = \frac{1}{2} \int_M \abs{F_A}^2 \, \mathrm{vol}_g.$$

## Remarks

Many authors omit the factor $\frac{1}{2}$; this does not change the critical points.
The functional is invariant under gauge transformations, since $F_{u^* A} = \operatorname{Ad}(u^{-1}) F_A$ and the inner product is invariant.
Its Euler–Lagrange equation is $d_A^* F_A = 0$, the [[Yang–Mills equations|JJWETW5AT8]].

Example: let $G = U(1)$ with Lie algebra $i\R$ and inner product $\ip{ia}{ib} = ab$, and let $P = M \times U(1)$ be the trivial bundle.
A connection is $A = d + i\alpha$ for a real $1$-form $\alpha$, and its curvature is $F_A = i \, d\alpha$.
So $\mathrm{YM}(A) = \frac{1}{2} \int_M \abs{d\alpha}^2 \, \mathrm{vol}_g$, the energy of the electromagnetic field with potential $\alpha$.
It vanishes exactly when $d\alpha = 0$, that is, when $A$ is flat.

Non-example: the integral $\int_M \tr(F_A \wedge F_A)$ for a matrix group $G$ on a $4$-manifold is not the Yang–Mills functional.
By Chern–Weil theory its value depends only on the bundle $P$ and not on $A$, so it has no variational content.

## Explanations

### L1 human

The functional measures the total squared curvature of a connection, so flat connections have the smallest possible value, zero.