Duistermaat–Heckman formula

Remarks

If $X$ is replaced by $iX$, the integral on the left becomes an oscillatory integral. The right side is then exactly its stationary phase approximation, so this approximation has no error. An equivalent form concerns the pushforward measure $\mu_*(\omega^n / n!)$, the Duistermaat–Heckman measure. It has a piecewise polynomial density with respect to Lebesgue measure on the affine span of $\mu(M)$. This form holds without the assumption of isolated fixed points. A modern proof uses equivariant cohomology and the Atiyah–Bott–Berline–Vergne localization formula.

Example: 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 $z$. By Archimedes, the area between heights $a < b$ is $2\pi(b - a)$, so the pushforward measure is $2\pi$ times Lebesgue measure on $[-1, 1]$. Then $\int_{S^2} e^{tz} \omega = 2\pi (e^t - e^{-t}) / t$, which matches the two fixed points $z = 1$ and $z = -1$ with weights $1$ and $-1$.

Going deeper

For a student

On $S^2$ the density is constant, which is Archimedes' theorem that slices of a sphere of equal height have equal area. The formula is an early instance of localization in equivariant cohomology, and it inspired many results on exact integration, including path integral heuristics in physics. Together with the Atiyah–Guillemin–Sternberg convexity theorem it gives a detailed picture of the image of a moment map.