Integrable dynamics in projective geometry via dimers and triple crossing diagram maps on the cylinder

TL;DR

This paper introduces twisted triple crossing diagram maps on the cylinder, providing geometric realizations of cluster integrable systems from bipartite graphs, and establishes connections between cross-ratio dynamics and cluster algebra structures.

Contribution

It defines new geometric maps related to bipartite graphs on cylinders and links them to known integrable systems, solving the open problem of cluster algebra structures for cross-ratio dynamics.

Findings

Geometric realization of cluster integrable systems from bipartite graphs.

Proof that pentagram map and cross-ratio dynamics are cluster integrable systems.

Identification of geometric R-matrices describing cross-ratio dynamics.

Abstract

We introduce twisted triple crossing diagram maps, collections of points in projective space associated to bipartite graphs on the cylinder, and use them to provide geometric realizations of the cluster integrable systems of Goncharov and Kenyon constructed from toric dimer models. Using this notion, we provide geometric proofs that the pentagram map and the cross-ratio dynamics integrable systems are cluster integrable systems. We show that in appropriate coordinates, cross-ratio dynamics is described by geometric $R$-matrices, which solves the open question of finding a cluster algebra structure describing cross-ratio dynamics.