Symplectic Reduction

Symplectic reduction (Marsden-Weinstein reduction) constructs lower-dimensional symplectic manifolds from Hamiltonian group actions. It is the symplectic counterpart of quotienting by symmetries and is fundamental in both physics and geometry.

Marsden-Weinstein Theorem

Definition4.7Symplectic Reduction

Let $(M, \omega)$ be a symplectic manifold with a Hamiltonian action of a Lie group $G$ and moment map $\mu: M \to \mathfrak{g}^*$ . For a regular value $\xi \in \mathfrak{g}^*$ with $G_\xi$ acting freely on $\mu^{-1}(\xi)$ , the symplectic reduction is $M_\xi = \mu^{-1}(\xi) / G_\xi$ , which carries a unique symplectic form $\omega_\xi$ satisfying $\pi^* \omega_\xi = \iota^* \omega$ .

Example$\mathbb{CP}^n$ as Reduction

$\mathbb{C}^{n+1}$ with the standard symplectic form $\omega = \frac{i}{2}\sum dz_k \wedge d\bar{z}_k$ has a Hamiltonian $S^1$ -action $e^{i\theta} \cdot z = e^{i\theta}z$ with moment map $\mu(z) = \frac{1}{2}|z|^2$ . The reduction at $\mu = 1/2$ gives $\mu^{-1}(1/2)/S^1 = S^{2n+1}/S^1 = \mathbb{CP}^n$ with the Fubini-Study symplectic form.

Shifting and Coadjoint Orbits

Definition4.8Coadjoint Orbit

A coadjoint orbit $\mathcal{O}_\xi = G \cdot \xi \subseteq \mathfrak{g}^*$ carries a natural symplectic structure, the Kirillov-Kostant-Souriau form: $\omega_\xi(\mathrm{ad}_X^* \xi, \mathrm{ad}_Y^* \xi) = \langle \xi, [X, Y] \rangle$ . Coadjoint orbits are the basic examples of symplectic manifolds with Hamiltonian group actions.

RemarkReduction in Stages

If $G = G_1 \times G_2$ acts on $(M, \omega)$ with moment map $\mu = (\mu_1, \mu_2)$ , then reducing by $G$ can be done in stages: first reduce by $G_1$ at $\mu_1 = \xi_1$ , then reduce the resulting symplectic manifold by the residual $G_2$ -action. This reduction in stages principle simplifies computations in many applications.