Symplectic quotient

If $\alpha$ is a regular value and the $G_\alpha$-action on $\mu^{-1}(\alpha)$ is free and proper, the orbit space is a quotient manifold. There is then a unique symplectic form $\omega_\alpha$ satisfying

where $i:\mu^{-1}(\alpha)\hookrightarrow M$ is inclusion and $\pi:\mu^{-1}(\alpha)\to M/\!/\!_\alpha G$ is the quotient map. This is the regular Marsden–Weinstein reduction theorem Ortega and Ratiu, Chapter 6.

Let $S^1$ act on $\mathbb C^n$ by scalar multiplication, with its standard symplectic form. Under the convention $d\langle\mu,\xi\rangle=\iota_{\xi_M}\omega$ used here, a moment map is $\mu(z)=-\tfrac12\lVert z\rVert^2$. For $\alpha<0$, the level set is a sphere and its quotient is $\mathbb{CP}^{n-1}$, carrying a scaled Fubini–Study form.

Regularity is not merely cosmetic. For a nonfree action or a critical value, orbit dimensions can jump, so the quotient need not be a manifold; singular symplectic reduction instead organizes it into symplectic strata Ortega and Ratiu, Chapters 8–9.

Some authors reserve “symplectic quotient” for a regular reduced manifold and use “reduced space” for the possibly singular orbit space. Reduction at the coadjoint orbit through $\alpha$ is also written $\mu^{-1}(\mathcal O_\alpha)/G$; under the standard hypotheses it corresponds to the level-set formulation above. The notation $M/\!/G$ usually means reduction at $0$, but the chosen value should always be stated.

1. Juan-Pablo Ortega and Tudor S. Ratiu, Momentum Maps and Hamiltonian Reduction, Birkhäuser, 2004. Springer DOI record. Relevant: Chapter 6 on regular reduction and Chapters 8–9 on singular reduction.
2. Victor Guillemin and Shlomo Sternberg, Symplectic Techniques in Physics, Cambridge University Press, 1984. Cambridge DOI record. Relevant: Chapter 5 on symplectic reduction.