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  "body": "\n## Definition\n\nLet $G = SU(n)$, a compact Lie group with Lie algebra $\\mathfrak{su}(n)$, and let $M$ be a closed oriented [[3-manifold|R20TB2KEQQ]].\nEvery [[principal $G$-bundle|8E37E74XVP]] over $M$ is trivial, so a [[connection|NJ21PN5ZGJ]] on it is given by a $\\mathfrak{su}(n)$-valued [[differential $1$-form|0E3G0GRPZ6]] $A$ on $M$.\nThe *Chern–Simons functional* is\n$$CS(A) = \\frac{1}{8\\pi^2} \\int_M \\tr\\left(A \\wedge dA + \\tfrac{2}{3} A \\wedge A \\wedge A\\right).$$\nFor an integer $k$, called the level, *Chern–Simons theory* is the gauge theory on $M$ with gauge group $G$ and action $S_k(A) = 2\\pi k \\, CS(A)$.\nIts quantum partition function is the formal path integral\n$$Z_k(M) = \\int e^{i S_k(A)} \\, \\mathcal{D}A$$\nover connections modulo gauge transformations.\n\n## Remarks\n\nExample: the trivial connection $A = 0$ has $CS(0) = 0$.\nThe first variation of $CS$ is $\\delta CS = \\frac{1}{4\\pi^2} \\int_M \\tr(\\delta A \\wedge F_A)$, where $F_A = dA + A \\wedge A$ is the curvature, so the critical points of $CS$ are exactly the flat connections.\n\n$CS$ is not invariant under gauge transformations: a gauge transformation $g \\colon M \\to G$ changes $CS(A)$ by an integer, the winding number of $g$ up to sign.\nSo $CS$ is not a function on connections modulo gauge, but $e^{i S_k}$ is, because $k$ is an integer; this is why the level must be an integer.\n\nThe path integral is not a rigorous mathematical definition.\nIts predicted values are realised rigorously by the Witten–Reshetikhin–Turaev invariants.\n",
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