Liouville measure
Let
be a symplectic manifold. The Liouville measure (or symplectic measure) of a Borel subset of is defined as
Duistermaat-Heckman Theorem
The Duistermaat-Heckman measure is a piecewise polynomial multiple of Lebesgue (or euclidean) measure
on , that is, the Radon-Nikodym derivative is piecewise polynomial. More specifically, for any Borel subset of , where is the Lebesgue volume form on and is polynomial on any region consisting of regular values of .
Symplectic Volume of Toric Symplectic Manifolds
In the case of a toric symplectic manifold, the Duistermaat-Heckman polynomial is a universal constant equal to
when its corresponding Delzant polytope is -dimensional. Thus the symplectic volume of is times the euclidean volume of .
e.g.
- For
with the standard toric symplectic structure, the moment map is given by and the moment polytope is the simplex with vertices and , , , . So the Euclidean volume of the simplex is . The symplectic volume of is then . - We can verify this in
.