We notice a remarkable connection between the Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four mutations. As an application of the new formalism, we show how to construct an integrable system with the spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into the double Bruhat cell of a Poisson-Lie group, show how triangular decomposition can be used to extend our approach to the general non-symmetric Newton polygons, and prove the Lemma which classifies conjugacy classes in double affine Weyl groups of A-type by decorated Newton polygons.

Solution of tetrahedron equation and cluster algebras / Gavrylenko, P.; Semenyakin, M.; Zenkevich, Y.. - In: JOURNAL OF HIGH ENERGY PHYSICS. - ISSN 1029-8479. - 2021:5(2021), pp. 1-34. [10.1007/JHEP05(2021)103]

Solution of tetrahedron equation and cluster algebras

Gavrylenko, P.;
2021-01-01

Abstract

We notice a remarkable connection between the Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four mutations. As an application of the new formalism, we show how to construct an integrable system with the spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into the double Bruhat cell of a Poisson-Lie group, show how triangular decomposition can be used to extend our approach to the general non-symmetric Newton polygons, and prove the Lemma which classifies conjugacy classes in double affine Weyl groups of A-type by decorated Newton polygons.
2021
2021
5
1
34
103
https://doi.org/10.1007/JHEP05(2021)103
https://arxiv.org/abs/2010.15871
Gavrylenko, P.; Semenyakin, M.; Zenkevich, Y.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/135598
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