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...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2025 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2025
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/213198 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Integrable Dynamics in Projective Geometry via Dimers and Triple Crossing Diagram Maps on the Cylinder. Niklas Christoph Affolter, Terrence George and Sanjay Ramassamy. SIGMA 21 (2025), 040, 48 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Summary: | 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 -matrices, which solves the open question of finding a cluster algebra structure describing cross-ratio dynamics.
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| ISSN: | 1815-0659 |