Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 040, 48 pages      arXiv:2108.12692      https://doi.org/10.3842/SIGMA.2025.040

Integrable Dynamics in Projective Geometry via Dimers and Triple Crossing Diagram Maps on the Cylinder

Niklas Christoph Affolter abc, Terrence George d and Sanjay Ramassamy e
a) Technische Universität Berlin, Institute of Mathematics, Strasse des 17. Juni 136, 10623 Berlin, Germany
b) Département de mathématiques et applications, École Normale Supérieure, CNRS, PSL University, 45 rue d'Ulm, 75005 Paris, France
c) Institute of Discrete Mathematics and Geometry, TU Wien, Wiedner Hauptstraße 8–10/104, 1040 Wien, Austria
d) Department of Mathematics, UCLA, 520 Portola Plaza, Los Angeles, CA 90095, USA
e) Université Paris-Saclay, CNRS, CEA, Institut de Physique Théorique, 91191 Gif-sur-Yvette, France

Received December 23, 2024, in final form May 20, 2025; Published online June 03, 2025

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.

Key words: discrete integrable systems; dimer model; cluster algebras; pentagram map; triple crossing diagram maps.

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