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  "body": "\n## Definition\n\nLet $(M, g)$ be an oriented [[Riemannian manifold|Q61VFHCH2J]] of dimension $n$.\nLet $G$ be a compact Lie group with Lie algebra $\\mathfrak{g}$, and let $P \\to M$ be a [[principal $G$-bundle|8E37E74XVP]].\nLet $A$ be a [[connection|NJ21PN5ZGJ]] on $P$ with [[curvature|ZAJFW1S078]] $F_A$, a $2$-form on $M$ with values in the adjoint bundle $\\mathrm{ad}\\, P = P \\times_G \\mathfrak{g}$.\nLet $d_A$ be the exterior [[covariant derivative|5FZDQQHQKM]] of $A$ on $\\mathrm{ad}\\, P$-valued forms, and let $\\star$ be the [[Hodge star operator|1Y4WX4XQWJ]] of $g$.\nThe connection $A$ satisfies the *Yang–Mills equations* if\n$$d_A \\star F_A = 0.$$\nSuch a connection is called a *Yang–Mills connection*.\n\n## Remarks\n\nThe equation is equivalent to $d_A^* F_A = 0$, where $d_A^*$ is the formal adjoint of $d_A$.\nEvery connection also satisfies the Bianchi identity $d_A F_A = 0$.\nFor compact $M$, the Yang–Mills connections are the critical points of the Yang–Mills functional $\\mathrm{YM}(A) = \\tfrac{1}{2} \\int_M \\abs{F_A}^2 \\, d\\mathrm{vol}_g$, where the norm uses an invariant inner product on $\\mathfrak{g}$.\n\nExample: a flat connection, with $F_A = 0$, satisfies the equations.\nOn a four-manifold, a connection with $\\star F_A = \\pm F_A$, an instanton or anti-instanton, also satisfies them, because $d_A \\star F_A = \\pm d_A F_A = 0$ by the Bianchi identity.\n\nNon-example: let $M = \\R^4$ with the Euclidean metric and $G = U(1)$, so that $d_A = d$, and take $A = i x_1^2 \\, dx_2$.\nThen $F_A = 2i x_1 \\, dx_1 \\wedge dx_2$, $\\star F_A = 2i x_1 \\, dx_3 \\wedge dx_4$, and $d \\star F_A = 2i \\, dx_1 \\wedge dx_3 \\wedge dx_4 \\ne 0$.\n\nFor $G = U(1)$ and a Lorentzian metric the equations are Maxwell's equations in vacuum.\n",
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