Potential functions on Grassmannians of planes and cluster transformations

TL;DR

This paper explores how potential functions on Grassmannians of planes, associated with polygon triangulations, are connected through cluster transformations, revealing their relation to wall-crossing formulas in Floer theory.

Contribution

It demonstrates that potential functions from different triangulations are related via cluster transformations and identifies these transformations with wall-crossing formulas in Floer theory.

Findings

Potential functions glue via cluster transformations.

Cluster transformations match wall-crossing formulas.

Provides a geometric interpretation of cluster transformations.

Abstract

With a triangulation of a planar polygon with $n$ sides, one can associate an integrable system on the Grassmannian of 2-planes in an $n$-space. In this paper, we show that the potential functions of Lagrangian torus fibers of the integrable systems associated with different triangulations glue together by cluster transformations. We also prove that the cluster transformations coincide with the wall-crossing formula in Lagrangian intersection Floer theory.