Toric degenerations of integrable systems on Grassmannians and polygon spaces

TL;DR

This paper constructs a new integrable system on Grassmannians linked to polygon triangulations, analyzes its moment polytopes, and explores their transformations and potential functions, advancing understanding of symplectic geometry and integrable systems.

Contribution

It introduces a novel integrable system on Grassmannians associated with polygon triangulations and studies the relations between their moment polytopes and potential functions.

Findings

Moment polytopes are related by integral piecewise-linear transformations.

Potential functions are connected via geometric lifts of these transformations.

The system provides new insights into the geometry of Grassmannians and polygon spaces.

Abstract

We introduce a completely integrable system on the Grassmannian of 2-planes in an n-space associated with any triangulation of a polygon with n sides, and compute the potential function for its Lagrangian torus fiber. The moment polytopes of this system for different triangulations are related by an integral piecewise-linear transformation, and the corresponding potential functions are related by its geometric lift in the sense of Berenstein and Zelevinsky.