Hearing Delzant polytopes from the equivariant spectrum

Let $M^{2n}$ be a symplectic toric manifold with a fixed $\mathbb{T}^n$-action and with a toric Kähler metric $g$. Abreu (2003) asked whether the spectrum of the Laplace operator $\Delta _g$ on $\mathcal {C}^\infty (M)$ 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 $M^4$ 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 $M_{\mathbb{R}}$ 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 $M$.