Cross-ratio dynamics and the dimer cluster integrable system
| Autor: | Affolter, Niklas, George, Terrence, Ramassamy, Sanjay |
|---|---|
| Přispěvatelé: | Ramassamy, Sanjay, Dimères : de la combinatoire à la mécanique quantique - - DIMERS2018 - ANR-18-CE40-0033 - AAPG2018 - VALID, Institut für Mathematik [Berlin], Technische Universität Berlin (TU), Department of Mathematics - University of Michigan, University of Michigan [Ann Arbor], University of Michigan System-University of Michigan System, Université Paris-Saclay, Centre National de la Recherche Scientifique (CNRS), CEA- Saclay (CEA), Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Institut de Physique Théorique - UMR CNRS 3681 (IPHT), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Deutsche Forschungsgemeinschaft (DFG) Collaborative Research Center TRR 109 'Discretization in Geometry and Dynamics', Bourse Tremplin@INP du CNRS, ANR-18-CE40-0033,DIMERS,Dimères : de la combinatoire à la mécanique quantique(2018), Technical University of Berlin / Technische Universität Berlin (TU) |
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Mathematics - Differential Geometry
[NLIN.NLIN-SI] Nonlinear Sciences [physics]/Exactly Solvable and Integrable Systems [nlin.SI] Nonlinear Sciences - Exactly Solvable and Integrable Systems [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] FOS: Physical sciences [MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS] Dynamical Systems (math.DS) [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO] Differential Geometry (math.DG) [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] FOS: Mathematics Mathematics - Combinatorics [NLIN.NLIN-SI]Nonlinear Sciences [physics]/Exactly Solvable and Integrable Systems [nlin.SI] Combinatorics (math.CO) Mathematics - Dynamical Systems Exactly Solvable and Integrable Systems (nlin.SI) [MATH.MATH-DG] Mathematics [math]/Differential Geometry [math.DG] |
| Popis: | Cross-ratio dynamics, allowing to construct 2D discrete conformal maps from 1D initial data, is a well-known discrete integrable system in discrete differential geometry. We relate it to the dimer integrable system from statistical mechanics by identifying its invariant Poisson structure and integrals of motion recently found by Arnold et al. to the Goncharov-Kenyon counterparts for the dimer model on a specific class of graphs. This solves the open question of finding a cluster algebra structure describing cross-ratio dynamics. The main tool relating geometry to the dimer model is the definition of triple crossing diagram maps associated to bipartite graphs on the cylinder. In passing we write the bivariate polynomial defining the dimer spectral curve for arbitrary bipartite graphs on the torus as the characteristic polynomial of a one-parameter family of matrices, a result which may be of independent interest. Comment: 30 pages, 11 figures |
| Databáze: | OpenAIRE |
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