Symplectic reduction

The aim of symplectic reduction is to find a way of taking quotients of symplectic manifolds under group actions. For example, suppose we have a free action of on a symplectic manifold . Naively, we might hope to find a symplectic structure on the topological quotient . However, this cannot possibly work, since

is odd, and symplectic manifolds always have even dimension.

Instead, we use the following trick: if the action of is Hamiltonian, then we can cut down the dimension by by restricting the action to a level set of the moment map. Taking the quotient of this new manifold, we at least get something even-dimensional. The following proposition ensures that we get a natural symplectic structure.

Proposition 6.1

Let be a symplectic manifold with a Hamiltonian action of a compact Lie group and associated equivariant moment map . If is a regular value of such that acts freely on , then is a symplectic manifold with symplectic structure induced by .


Definition 6.9

The symplectic manifold of the previous Proposition is called the symplectic reduction of , and is denoted by if the choice moment map is understood.


Remark 6.7

If is any equivariant moment map and is invariant under the coadjoint action of , then defines another moment map. (In particular if is abelian, then we can do this for any .) If is a regular value for such that the -action on is free, then we can form another symplectic reduction

In general, different values of will give different symplectic reductions. We often write
where the choice of moment map is implicit.


Proof (Sketch of proof of Proposition before)

Write . Since the -action on is free, is a manifold with tangent space

where is any preimage of , and are the components for the moment map on corresponding to some basis for . Define the symplectic form by
where is a preimage of , and are preimages of . The form is well-defined since is invariant under and the are constant restricted to (so that for all ). One can check that is indeed a symplectic form.


Example 6.9 ($\mathbb{CP}^n$)

Consider acting on diagonally by

The moment map (which is just a Hamiltonian in this case) is
So for every , we get a symplectic structure on
The associated symplectic form is the unique form such that
where is the quotient map.


Exercise 6.7

Show that for an appropriate choice of , agrees with the explicit expression for the Fubini-Study form in the lecture on Kähler geometry.


Example 6.10

The -action of Example before factors through the -action

via the diagonal map
So we get a residual action of on the symplectic reduction with moment map
induced by the moment map
for the -action. Note that the image of is contained in
which, up to translation, is the same as
So does make sense as a moment map. For example, the moment image of is the interval shown below.

Tik figure 2.png

The moment image of is the triangle below.

Tik figure 3 version 2.png

In general, the moment image of is the -simplex