Moment map

Summary

A moment map, also called a momentum map, packages the conserved quantities of a symmetry of a symplectic manifold into one map. When a Lie group $G$ acts on $(M, \omega)$, a moment map is a map $\mu \colon M \to \mathfrak{g}^*$ whose components generate the action as Hamiltonian vector fields. It is the main tool for symplectic reduction, for convexity theorems and for toric geometry.