Proof of Marsden-Weinstein Reduction

We prove that the symplectic reduction $M_\xi = \mu^{-1}(\xi)/G_\xi$ carries a natural symplectic structure induced from the ambient manifold.

Statement

Theorem4.3Marsden-Weinstein Reduction

Let $(M, \omega)$ be a symplectic manifold with a free Hamiltonian action of a Lie group $G$ and equivariant moment map $\mu: M \to \mathfrak{g}^*$ . If $\xi \in \mathfrak{g}^*$ is a regular value of $\mu$ , then $M_\xi = \mu^{-1}(\xi)/G_\xi$ is a smooth manifold of dimension $\dim M - 2\dim G_\xi$ , and there is a unique symplectic form $\omega_\xi$ on $M_\xi$ satisfying $\pi^*\omega_\xi = \iota^*\omega$ , where $\iota: \mu^{-1}(\xi) \hookrightarrow M$ and $\pi: \mu^{-1}(\xi) \to M_\xi$ .

Proof

Step 1: $\mu^{-1}(\xi)$ is a smooth submanifold. Since $\xi$ is a regular value of $\mu$ , $\mu^{-1}(\xi)$ is a smooth submanifold of $M$ with $\dim \mu^{-1}(\xi) = \dim M - \dim G$ .

Step 2: $G_\xi$ acts freely. The stabilizer $G_\xi = \{g \in G : \mathrm{Ad}_g^* \xi = \xi\}$ preserves $\mu^{-1}(\xi)$ by equivariance. Since the action is free, $M_\xi$ is a smooth manifold.

Step 3: Symplectic form on the quotient. At $x \in \mu^{-1}(\xi)$ , the tangent space splits as $T_x M = T_x(\mu^{-1}(\xi)) + T_x(G \cdot x)$ . The kernel of $\iota^*\omega|_{T_x(\mu^{-1}(\xi))}$ is precisely $T_x(G_\xi \cdot x)$ .

Indeed, for $v \in T_x(\mu^{-1}(\xi))$ and $X_\eta =$ infinitesimal generator of $\eta \in \mathfrak{g}_\xi$ : $\omega(v, X_\eta) = d\mu_\eta(v) = 0$ since $v$ is tangent to $\mu^{-1}(\xi)$ . This shows $T_x(G_\xi \cdot x) \subseteq \ker(\iota^*\omega)$ . A dimension count gives equality.

Step 4: Non-degeneracy. Since $\ker(\iota^*\omega) = T_x(G_\xi \cdot x)$ , the form $\iota^*\omega$ descends to a well-defined non-degenerate 2-form $\omega_\xi$ on $M_\xi$ . Closedness of $\omega_\xi$ follows from $d(\iota^*\omega) = \iota^*(d\omega) = 0$ . $\square$

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ExampleToric Manifolds

A toric manifold is a $2n$ -dimensional compact symplectic manifold with an effective Hamiltonian $T^n$ -action. The moment map image is a convex polytope $\Delta \subseteq \mathbb{R}^n$ (by the Atiyah-Guillemin-Sternberg convexity theorem). The toric manifold is reconstructed from $\Delta$ via iterated symplectic reduction. The Delzant theorem classifies toric manifolds by their moment polytopes.

RemarkSingular Reduction

When $G$ does not act freely on $\mu^{-1}(\xi)$ , the quotient is a stratified symplectic space. Sjamaar and Lerman developed the theory of singular symplectic reduction, showing that each stratum is a symplectic manifold and the strata fit together in a controlled way.