#### Title

Hearing Delzant Polytopes From the Equivariant Spectrum

#### Publication Date

2012

#### Description

Let M^{2n} be a symplectic toric manifold with a fixed T^n-action and with a toric K\"ahler metric g. Abreu 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_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.

#### Journal

Transactions of the American Mathematical Society

#### Volume

364

#### Issue

2

#### First Page

887

#### Last Page

910

#### Department

Mathematics

#### Recommended Citation

Dryden, Emily; Guillemin, Victor; and Sena-Dias, Rosa. "Hearing Delzant Polytopes From the Equivariant Spectrum." *Transactions of the American Mathematical Society* (2012)
: 887-910.