Potential functions on Grassmannians of planes and cluster transformations

2017/11/13 by Yuichi Nohara, Nohara, Yuichi, Kazushi Ueda +1
Mathematics · #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Symplectic Geometry (math.SG) #math.SG

paper · pdf · doi:10.48550/arxiv.1711.04456

47 pages. v2: revised following referee's comments

openalex publication_date 2017/11/13 · arxiv created 2019/02/03 · arxiv updated 2019/02/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Mirror symmetry and T-duality in the complement of an anticanonical divisor

Related

• Disk potential functions for polygon spaces
• The wall-crossing formula and Lagrangian mutations
• Quivers with potentials associated to triangulated surfaces
• Cluster Duality for Lagrangian and Orthogonal Grassmannians
• Integrable dynamics in projective geometry via dimers and triple crossing diagram maps on the cylinder