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  "body": "\n## Summary\n\nA Chern–Simons form is an odd-degree differential form built from a connection whose exterior derivative is a characteristic form of Chern–Weil theory.\nThe best-known one is the $3$-form $\\mathrm{tr}(A \\wedge dA + \\frac23 A \\wedge A \\wedge A)$, whose derivative is $\\mathrm{tr}(F \\wedge F)$.\nChern–Simons forms give secondary invariants, defined when the primary characteristic forms vanish.\nTheir integrals over $3$-manifolds are the action of Chern–Simons gauge theory, which is central in low-dimensional topology and physics.\n\n## Explanations\n\n### L1 human\n\nThe characteristic forms of [[Chern–Weil theory|82ZPRD39J7]] are closed, and on a trivial bundle they are even exact.\nA Chern–Simons form is an explicit form whose derivative is such a characteristic form.\nIt is like an antiderivative of the curvature expression.\nOn a closed $3$-manifold the integral of the Chern–Simons $3$-form is a number that changes in a controlled way under gauge transformations.\nThis number is used to define invariants of knots and of $3$-manifolds.\n",
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