Journal of Symplectic Geometry

Volume 18 (2020)

Number 2

Potential functions on Grassmannians of planes and cluster transformations

Pages: 559 – 612



Yuichi Nohara (Department of Mathematics, School of Science and Technology, Meiji University, Kawasaki-shi, Kanagawa, Japan)

Kazushi Ueda (Graduate School of Mathematical Sciences, University of Tokyo, Meguro-ku, Tokyo, Japan)


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.

Received 14 February 2018

Accepted 1 February 2019

Published 8 June 2020