Delzant's theorem explained

In mathematics, a Delzant polytope is a convex polytope in

R ^{n} such for each vertex v , exactly n edges meet at v (that is, it is a simple polytope), and these edges form a collection of vectors that form a Z -basis of Z ^{n} . Delzant's theorem, introduced by, classifies effective Hamiltonian torus actions on compact connected symplectic manifolds by the image of the associated moment map, which is a Delzant polytope.

The theorem states that there is a bijective correspondence between symplectic toric manifolds (up to torus-equivariant symplectomorphism) and Delzant polytopes -- more precisely, the moment polytope of a symplectic toric manifold is a Delzant polytope, every Delzant polytope is the moment polytope of such a manifold, and any two such manifolds with the equivalent moment polytopes (up to translations) admit a torus-equivariant symplectomorphism between them.