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  "body": "\n## Definition\n\nLet $M$ be a smooth manifold, and let $\\mathfrak{gl}_n(\\C)$ be the space of complex $n \\times n$ matrices.\nLet $A$ be a smooth $\\mathfrak{gl}_n(\\C)$-valued [[$1$-form|0E3G0GRPZ6]] on $M$; it is a connection form on the trivial bundle $M \\times \\C^n$.\nLet $F = dA + A \\wedge A$ be its [[curvature|ZAJFW1S078]], where $d$ is the [[exterior derivative|0ZHSTAG8ZY]] and $\\wedge$ is the matrix [[wedge product|0GRAATMGP1]].\nThe *Chern–Simons $3$-form* of $A$ is\n$$\\mathrm{CS}(A) = \\tr\\Big(A \\wedge dA + \\tfrac23 \\, A \\wedge A \\wedge A\\Big).$$\n\n## Remarks\n\nThe defining property is $d\\, \\mathrm{CS}(A) = \\tr(F \\wedge F)$.\nCheck: $d \\tr(A \\wedge dA) = \\tr(dA \\wedge dA)$ and $d \\tr(A \\wedge A \\wedge A) = 3 \\tr(dA \\wedge A \\wedge A)$ by cyclicity of the graded trace; while $\\tr(F \\wedge F) = \\tr(dA \\wedge dA) + 2 \\tr(dA \\wedge A \\wedge A) + \\tr(A^4)$ and $\\tr(A^4) = 0$.\nExample: if $n = 1$, then $A \\wedge A = 0$, $F = dA$ and $\\mathrm{CS}(A) = A \\wedge dA$; on $M = \\R^4$ with coordinates $x_1, \\dots, x_4$ and $A = x_1 \\, dx_2 + x_3 \\, dx_4$, one gets $dA = dx_1 \\wedge dx_2 + dx_3 \\wedge dx_4$, and both $d(A \\wedge dA)$ and $F \\wedge F$ equal $2 \\, dx_1 \\wedge dx_2 \\wedge dx_3 \\wedge dx_4$.\nNon-example: for non-abelian $A$, the form $\\tr(A \\wedge dA)$ alone does not have derivative $\\tr(F \\wedge F)$ in general, since the term $2 \\tr(dA \\wedge A \\wedge A)$ is missing; the cubic term is needed.\nMore generally, let $f$ be an invariant polynomial of degree $k$ as in the [[Chern–Weil homomorphism|82ZPRD39J7]], and let $\\omega$ be a connection on a principal bundle $P$.\nPut $\\Omega_t = t\\, d\\omega + \\frac{t^2}{2} [\\omega \\wedge \\omega]$ and $Tf(\\omega) = k \\int_0^1 f(\\omega, \\Omega_t, \\dots, \\Omega_t) \\, dt$.\nThen $Tf(\\omega)$ is a $(2k-1)$-form on $P$ with $d \\, Tf(\\omega) = f(\\Omega, \\dots, \\Omega)$.\nThe $3$-form above is the case $f(X, Y) = \\tr(XY)$ in a trivialization.\nUnder a gauge transformation $A \\mapsto g^{-1} A g + g^{-1} dg$, $\\mathrm{CS}(A)$ changes by an exact form plus $-\\frac13 \\tr\\big((g^{-1}dg)^3\\big)$, whose integral over a closed $3$-manifold is an integer multiple of $8\\pi^2$ for $G = \\mathrm{SU}(n)$.\n",
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