From Lagrangian Products to Toric Domains via the Toda Lattice

TL;DR

This paper demonstrates that specific Lagrangian product configurations in symplectic geometry are equivalent to toric domains using the Toda lattice, including a case where a product of a triangle and hexagon is symplectomorphic to a 4D Euclidean ball.

Contribution

It establishes a novel connection between Lagrangian products and toric domains via the Toda lattice, providing explicit symplectic equivalences including a new example involving a triangle and hexagon.

Findings

Lagrangian products of certain polytopes are symplectically equivalent to toric domains.

The product of a simplex and Voronoi cell of $A_n$ is equivalent to a Euclidean ball.

The product of an equilateral triangle and a regular hexagon is symplectomorphic to a 4D Euclidean ball.

Abstract

In this paper we use the periodic Toda lattice to show that certain Lagrangian product configurations in the classical phase space are symplectically equivalent to toric domains. In particular, we prove that the Lagrangian product of a certain simplex and the Voronoi cell of the root lattice $A_n$ is symplectically equivalent to a Euclidean ball. As a consequence, we deduce that the Lagrangian product of an equilateral triangle and a regular hexagon is symplectomorphic to a Euclidean ball in dimension 4.