Cluster Reductions, Mutations, and $q$-Painlevé Equations
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| Format: | Preprint |
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2024
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| _version_ | 1866910680466063360 |
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| author | Bershtein, Mikhail Gavrylenko, Pavlo Marshakov, Andrei Semenyakin, Mykola |
| author_facet | Bershtein, Mikhail Gavrylenko, Pavlo Marshakov, Andrei Semenyakin, Mykola |
| contents | We propose an extension of the Goncharov-Kenyon class of cluster integrable systems by their Hamiltonian reductions. This extension allows us to fill in the gap in cluster construction of the $q$-difference Painlevé equations, showing that all of them can be obtained as deautonomizations of the reduced Goncharov-Kenyon systems.
Conjecturally, the isomorphisms of reduced Goncharov-Kenyon integrable systems are given by mutations in another, dual in some sense, cluster structure. These are the polynomial mutations of the spectral curve equations and polygon mutations of the corresponding decorated Newton polygons. In the Painlevé case the initial and dual cluster structures are isomorphic. It leads to self-duality between the spectral curve equation and the Painlevé Hamiltonian, and also extends the symmetry from affine to elliptic Weyl group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_00325 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cluster Reductions, Mutations, and $q$-Painlevé Equations Bershtein, Mikhail Gavrylenko, Pavlo Marshakov, Andrei Semenyakin, Mykola Exactly Solvable and Integrable Systems High Energy Physics - Theory Mathematical Physics Quantum Algebra We propose an extension of the Goncharov-Kenyon class of cluster integrable systems by their Hamiltonian reductions. This extension allows us to fill in the gap in cluster construction of the $q$-difference Painlevé equations, showing that all of them can be obtained as deautonomizations of the reduced Goncharov-Kenyon systems. Conjecturally, the isomorphisms of reduced Goncharov-Kenyon integrable systems are given by mutations in another, dual in some sense, cluster structure. These are the polynomial mutations of the spectral curve equations and polygon mutations of the corresponding decorated Newton polygons. In the Painlevé case the initial and dual cluster structures are isomorphic. It leads to self-duality between the spectral curve equation and the Painlevé Hamiltonian, and also extends the symmetry from affine to elliptic Weyl group. |
| title | Cluster Reductions, Mutations, and $q$-Painlevé Equations |
| topic | Exactly Solvable and Integrable Systems High Energy Physics - Theory Mathematical Physics Quantum Algebra |
| url | https://arxiv.org/abs/2411.00325 |