Proof of the Duistermaat-Heckman Theorem

The Duistermaat-Heckman theorem describes the exact behavior of the push-forward of the symplectic volume form under the moment map. It shows that this measure is piecewise polynomial, a remarkable exactness result.

Statement

Theorem5.3Duistermaat-Heckman Theorem

Let $(M^{2n}, \omega)$ be a compact symplectic manifold with a Hamiltonian $S^1$ -action and moment map $\mu: M \to \mathbb{R}$ . Then the Duistermaat-Heckman measure $\mu_*\!\left(\frac{\omega^n}{n!}\right)$ is a piecewise polynomial measure on $\mathbb{R}$ . Equivalently, the oscillatory integral $\int_M e^{i t \mu} \frac{\omega^n}{n!} = \sum_{F \subseteq M^{S^1}} \frac{e^{it\mu(F)}}{\prod_{j=1}^{n_F} (i t w_j(F))}$ is computed exactly by the fixed-point contributions (localization formula), where $w_j(F)$ are the weights of the $S^1$ -action on the normal bundle of the fixed component $F$ .

Proof Sketch

Proof

The key tool is the equivariant cohomology localization formula (Atiyah-Bott-Berline-Vergne).

The equivariant symplectic form $\omega + t\mu$ is equivariantly closed. The integral $\int_M e^{\omega + t\mu}$ can be computed by localization: $\int_M e^{\omega + t\mu} = \sum_{F} \int_F \frac{e^{\omega_F + t\mu(F)}}{e_T(N_F)},$ where the sum is over fixed components $F$ , $\omega_F$ is the restriction of $\omega$ , and $e_T(N_F) = \prod_j (tw_j)$ is the equivariant Euler class of the normal bundle.

For isolated fixed points $p$ with weights $w_1, \ldots, w_n$ : $\int_M e^{\omega + t\mu} = \sum_p \frac{e^{t\mu(p)}}{\prod_{j=1}^n (tw_j(p))}.$

The Duistermaat-Heckman measure is the inverse Fourier transform of this expression, which is piecewise polynomial. $\square$

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Example$S^2$ with Rotation

For $S^2$ with the standard $S^1$ -rotation and moment map $\mu =$ height function, the DH measure is $\mu_*(\omega) = dt$ on $[-1, 1]$ (the uniform measure). The fixed points are the north and south poles with $\mu = \pm 1$ and weights $\pm 1$ : $\int_{S^2} e^{it\mu}\omega = \frac{e^{it}}{it} + \frac{e^{-it}}{-it} = \frac{2\sin t}{t}.$

RemarkWall-Crossing

When $\mu$ crosses a critical value, the DH polynomial changes. The change (wall-crossing formula) is determined by the fixed-point data at the critical level. This connects the DH theorem to the variation of symplectic quotients.