Integrable Dynamics in Projective Geometry via Dimers and Triple Crossing Diagram Maps on the Cylinder
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.
Authors
- Niklas Affolter (ORCID: https://orcid.org/0000-0003-3936-2060)
- Terrence George (ORCID: https://orcid.org/0000-0001-8199-3136)
- Sanjay Ramassamy (ORCID: https://orcid.org/0000-0003-0746-9912)
Institutions
- Centre National de la Recherche Scientifique (FR)
- Commissariat à l'Énergie Atomique et aux Énergies Alternatives (FR)
- Université Paris-Saclay (FR)
- University of Michigan (US)
- École Normale Supérieure - PSL (FR)
- Laboratoire de Mathématiques d'Orsay (FR)
- Département de mathématiques et applications (FR)
- CEA Paris-Saclay (FR)
- Institut de Physique Théorique (FR)
- Technische Universität Berlin (DE)
Publication Details
- Journal
- Symmetry Integrability and Geometry Methods and Applications
- Published
- 2025-06-03
- DOI
- https://doi.org/10.3842/sigma.2025.040
- Citations
- 4
- Primary Topic
- Control and Dynamics of Mobile Robots
- Type
- article
- Field-Weighted Citation Impact
- 1.21
Funders
- Deutsche Forschungsgemeinschaft
- Agence Nationale de la Recherche
- Technische Universität Wien
- Centre National de la Recherche Scientifique