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  "body": "\n## Statement\n\nLet $(M, \\omega)$ be a compact [[symplectic manifold|CAN490EPRT]] of dimension $2n$.\nLet a torus $T$ act on $M$ with a [[moment map|ZARM34FZ6Z]] $\\mu \\colon M \\to \\mathfrak{t}^*$, and assume that the fixed points of $T$ are isolated.\nAt a fixed point $p$, identify $(T_p M, \\omega_p)$ with $\\C^n$ by an $\\omega_p$-compatible complex structure.\nLet $w_{1,p}, \\ldots, w_{n,p} \\in \\mathfrak{t}^*$ be the weights of the action of $T$ on it, with signs fixed by the same convention as the sign of $\\mu$.\nThen for every $X \\in \\mathfrak{t}$ with $\\langle w_{j,p}, X \\rangle \\ne 0$ for all $j$ and $p$,\n$$\\int_M e^{\\langle \\mu, X \\rangle} \\frac{\\omega^n}{n!} = (2\\pi)^n \\sum_{p \\in M^T} \\frac{e^{\\langle \\mu(p), X \\rangle}}{\\prod_{j=1}^n \\langle w_{j,p}, X \\rangle}.$$\n\n## Remarks\n\nIf $X$ is replaced by $iX$, the integral on the left becomes an oscillatory integral.\nThe right side is then exactly its stationary phase approximation, so this approximation has no error.\nAn equivalent form concerns the [[pushforward measure|3N66TM3TSQ]] $\\mu_*(\\omega^n / n!)$, the Duistermaat–Heckman measure.\nIt has a piecewise polynomial density with respect to [[Lebesgue measure|2Z2QVWZ262]] on the affine span of $\\mu(M)$.\nThis form holds without the assumption of isolated fixed points.\nA modern proof uses equivariant cohomology and the Atiyah–Bott–Berline–Vergne localization formula.\n\nExample: the circle acts on the unit sphere $S^2$ by rotation about the vertical axis, $\\omega$ is the area form of total area $4\\pi$, and $\\mu$ is the height $z$.\nBy Archimedes, the area between heights $a \u003c b$ is $2\\pi(b - a)$, so the pushforward measure is $2\\pi$ times Lebesgue measure on $[-1, 1]$.\nThen $\\int_{S^2} e^{tz} \\omega = 2\\pi (e^t - e^{-t}) / t$, which matches the two fixed points $z = 1$ and $z = -1$ with weights $1$ and $-1$.\n\n## Explanations\n\n### L1 human\n\nWhen a torus symmetry acts on a compact symplectic manifold, certain integrals over the whole manifold can be computed from the few points that the symmetry fixes.\nThe formula replaces an integral by a finite sum.\nIt also says that the symplectic volume is spread over the moment polytope by a piecewise polynomial density.\n\n### L2 human\n\nOn $S^2$ the density is constant, which is Archimedes' theorem that slices of a sphere of equal height have equal area.\nThe formula is an early instance of localization in equivariant cohomology, and it inspired many results on exact integration, including path integral heuristics in physics.\nTogether with the [[Atiyah–Guillemin–Sternberg convexity theorem|5CTHC6612N]] it gives a detailed picture of the image of a moment map.\n",
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