Delzant's theorem

In mathematics, a Delzant polytope is a convex polytope in $$\mathbb{R}^n$$ 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 $$\mathbb{Z}$$-basis of $$\mathbb{Z}^n$$. 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.