Symplectic Reduction

Hamiltonian Actions

Symplectic Group Action

Suppose is a symplectic manifold. A Lie group action on on is symplectic if it acts on via symplectomorphisms.

Hamiltonian Action

Suppose is a Lie group with Lie algebra . A symplectic -action on is Hamiltonian if there exists a map such that

1. For all , the fundamental vector field is the Hamiltonian vector field generated by , where is given by
2. is -equivariant with respect to the given action of on and the coadjoint action of on :

is then called a hamiltonian -space and is a moment map.

e.g.

• The moment map of the Hamiltonian action of or on , is a function , which plays the role of a Hamiltonian function.
• For the case where is an -torus, the coadjoint action is trivial because . So the action is Hamiltonian when each restriction is Hamiltonian in the previous sense. That is, For each basis vector of , the th component of is a hamiltonian function that is invariant under the action of the torus. Moreover, if is a moment map for a torus action, then clearly any of its translations is also a moment map for that action. Reciprocally, any two moment maps for a given hamiltonian torus action differ by a constant. We will investigate the torus action further in the context of toric symplectic manifolds.

Symplectic Reduction

Classical physicists realized that, whenever there is a symmetry group of dimension acting on a mechanical system, then the number of degrees of freedom for the position and momenta of the particles may be reduced by . Symplectic reduction formulates this feature mathematically.

Marsden-Weinstein-Meyer Theorem

Let be a Hamiltonian -space for a compact Lie group . Let be the inclusion map. Assume that acts freely on . Then

1. the orbit space is a manifold,
2. is a principal -bundle, and
3. there is a symplectic form on satisfying .

The pair is called the symplectic reduction of with respect to and .