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  "body": "\n## Definition\n\nLet $(M, g)$ be a compact oriented [[Riemannian manifold|7CBT4SM83Y]] without boundary, with volume form $\\mathrm{vol}_g$.\nLet $G$ be a compact [[Lie group|7JBRGCVJRB]] with Lie algebra $\\mathfrak{g}$.\nFix an inner product on $\\mathfrak{g}$ that is invariant under the adjoint action of $G$.\nLet $P \\to M$ be a [[principal $G$-bundle|8E37E74XVP]], and let $\\operatorname{ad} P = P \\times_G \\mathfrak{g}$ be its adjoint bundle.\nFor a [[connection|NJ21PN5ZGJ]] $A$ on $P$, let $F_A$ be its [[curvature|ZAJFW1S078]], a $2$-form on $M$ with values in $\\operatorname{ad} P$.\nThe metric $g$ and the inner product on $\\mathfrak{g}$ give a pointwise norm $\\abs{F_A}$.\nThe *Yang–Mills functional* is the map from the space of connections on $P$ to $[0, \\infty)$ given by\n$$\\mathrm{YM}(A) = \\frac{1}{2} \\int_M \\abs{F_A}^2 \\, \\mathrm{vol}_g.$$\n\n## Remarks\n\nMany authors omit the factor $\\frac{1}{2}$; this does not change the critical points.\nThe functional is invariant under gauge transformations, since $F_{u^* A} = \\operatorname{Ad}(u^{-1}) F_A$ and the inner product is invariant.\nIts Euler–Lagrange equation is $d_A^* F_A = 0$, the [[Yang–Mills equations|JJWETW5AT8]].\n\nExample: 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.\nA connection is $A = d + i\\alpha$ for a real $1$-form $\\alpha$, and its curvature is $F_A = i \\, d\\alpha$.\nSo $\\mathrm{YM}(A) = \\frac{1}{2} \\int_M \\abs{d\\alpha}^2 \\, \\mathrm{vol}_g$, the energy of the electromagnetic field with potential $\\alpha$.\nIt vanishes exactly when $d\\alpha = 0$, that is, when $A$ is flat.\n\nNon-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.\nBy Chern–Weil theory its value depends only on the bundle $P$ and not on $A$, so it has no variational content.\n\n## Explanations\n\n### L1 human\n\nThe functional measures the total squared curvature of a connection, so flat connections have the smallest possible value, zero.\n",
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