Toric Symplectic Manifold

Toric Actions

Among all kinds of symplectic reductions, the most well-studied are those that arise from toric symplectic manifolds. These manifolds are equipped with a Hamiltonian action of a torus. We shall study this before we proceed to the general case of symplectic reduction.

Toric Symplectic Manifold

A toric symplectic manifold is a is a symplectic manifold of dimension equipped with an effective Hamiltonian action of a torus , and with a choice of a corresponding moment map .

e.g.

• Consider acting on by coordinate-wise multiplication, i.e. acts on by The moment map is then given by up to a constant. We shall check this is a valid effective Hamiltonian action. It is clearly effective, only the identity acts trivially. By the example, we know the fundamental vector field generated by a basis vector of the Lie algebra is given by . Thus the corresponding Hamilton’s equations are given by It follows that up to a constant, which is the moment map we defined above.

Moment Polytope

The moment polytope of a toric symplectic manifold is its image under the moment map.

e.g.

• The circle acts on by rotations: with moment map equal to the height function and moment polytope .

Atiyah-Guillemin-Sternberg Convexity Theorem

Let be a compact connected symplectic manifold, and let be an -torus. Suppose that is a hamiltonian action with moment map . Then:

• the levels of are connected;
• the moment polytope is convex;
• the moment polytope is the convex hull of the images of the fixed points of the action.

Equivalence of Symplectic Toric Manifolds

Two symplectic toric manifolds, and are equivalent if there exists an isomorphism and a -equivariant symplectomorphism such that .

Delzant’s Theorem

Delzant Polytope

A Delzant polytope is a convex polytope in satisfying the following conditions:

• Simple: each vertex is the intersection of exactly edges.
• Rational: each edge meeting at the vertex is of the form , for , and fixed ;
• Smooth: for each vertex, the corresponding vectors can be chosen to be a basis of .

Delzant’s Theorem

There is a one-to-one correspondence between the equivalence classes of toric symplectic manifolds and Delzant polytopes. Specifically, this bijective correspondence is given by the moment map.

e.g.

Proposition

Let be an -dimensional Delzant polytope, and let be the associated symplectic toric manifold. Then maps the fixed points of bijectively onto the vertices of .

Blow-Ups of Toric Symplectic Manifolds

We start with the blow-up at origin of :

Blow-Up of

Let be the tautological line bundle over , i.e., a complex line bundle whose fiber over each point is precisely the line represented by that point. The blow-up of at origin is the total space of the bundle , that is, The corresponding blow-down map is The set is called the exceptional divisor.

Theorem

Suppose is an -dimensional Delzant polytope, and let be the associated symplectic toric manifold. The -blow-up of at a fixed point (i.e. is in the isotropy) is the symplectic toric manifold associated to the polytope obtained from by replacing the vertex by the vertices

Geometric Interpretation of Blow-Up

In other words, the moment polytope for the blow-up of at is obtained from by chopping off the corner corresponding to , thus substituting the original set of vertices by the same set with the vertex corresponding to replaced by exactly new vertices:

e.g. is a toric symplectic manifold with the standard action. It corresponds to the Delzant polytope . Chopping off the vertex yields the same polytope , so the blow-up manifold remains to be . One can see this through symplectic cut as well (refer to the example).

Symplectic Cuts

For toric symplectic manifolds, blow-ups are equivalent to symplectic cuts, which are a more general construction applicable to any symplectic manifold. We shall see this on with the standard action:

e.g. Let us apply the symplectic cut to at the origin point. Let be defined on . Then for some , we have Let and , then . Note that , , so

References and Other Resources

• Lerman, Symplectic Cuts