Delzant's theorem

Template:Short description In mathematics, a Delzant polytope is a convex polytope in ${ℝ}^n$ such that for each vertex $v$, exactly $n$ edges meet at $v$ (that is, it is a simple polytope), and there are integer vectors parallel to these edges forming a $\mathbb{Z}$-basis of ${\mathbb{Z}}^n$.

Delzant's theorem, introduced by Template:Harvs, 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 every symplectic toric manifold is a Delzant polytope, every Delzant polytope is the moment polytope of such a manifold, and any two such manifolds with equivalent moment polytopes (up to translations and $GL(n,\mathbb{Z})$ transformations) admit a torus-equivariant symplectomorphism between them.

References

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