Hearing Delzant polytopes from the equivariant spectrum

Abstract

Let M[superscript 2n] be a symplectic toric manifold with a fixed T[superscript n]-action and with a toric Kähler metric g. Abreu (2003) asked whether the spectrum of the Laplace operator Δ[subscript g] on C∞ (M) determines the moment polytope of M, and hence by Delzant's theorem determines M up to symplectomorphism. We report on some progress made on an equivariant version of this conjecture. If the moment polygon of M[superscript 4] is generic and does not have too many pairs of parallel sides, the so-called equivariant spectrum of M and the spectrum of its associated real manifold M[subscript R] determine its polygon, up to translation and a small number of choices. For M of arbitrary even dimension and with integer cohomology class, the equivariant spectrum of the Laplacian acting on sections of a naturally associated line bundle determines the moment polytope of M.