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  "body": "\n## Definition\n\nLet $(X, g)$ be an [[oriented|913WEQERBE]] [[Riemannian manifold|7CBT4SM83Y]] of dimension $4$, and let $\\star$ be its [[Hodge star|1Y4WX4XQWJ]], which maps 2-forms to 2-forms.\nLet $G$ be a compact Lie group with Lie algebra $\\mathfrak{g}$, and let $P \\to X$ be a [[principal|8E37E74XVP]] $G$-bundle.\nLet $A$ be a [[connection|NJ21PN5ZGJ]] on $P$ with [[curvature|ZAJFW1S078]] $F_A$, a 2-form on $X$ with values in the adjoint bundle $P \\times_G \\mathfrak{g}$.\nThe connection $A$ is an *instanton*, or anti-self-dual connection, if\n$$\\star F_A = -F_A.$$\n\n## Remarks\n\nSome authors use the opposite orientation and call self-dual connections, with $\\star F_A = F_A$, instantons.\nAnti-self-dual connections satisfy the Yang–Mills equations $d_A \\star F_A = 0$, because $d_A \\star F_A = -d_A F_A = 0$ by the Bianchi identity.\nOn a compact $X$, they minimize the Yang–Mills functional $\\int_X \\abs{F_A}^2$ among connections on $P$, the minimum being a topological constant times the absolute value of a characteristic number of $P$.\n\nExample: a flat connection, with $F_A = 0$, is an instanton.\nThe BPST instanton on $\\R^4$ for $G = SU(2)$ has curvature that is nowhere zero and anti-self-dual for one choice of orientation.\n\nNon-example: on the trivial $U(1)$-bundle over $\\R^4$ with coordinates $x_1, \\dots, x_4$, let $A = i \\, x_1 \\, dx_2$.\nThen $F_A = i \\, dx_1 \\wedge dx_2$ and $\\star F_A = i \\, dx_3 \\wedge dx_4 \\ne -F_A$, so $A$ is not an instanton.\n",
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