Solution of tetrahedron equation and cluster algebras

TL;DR

This paper establishes a novel link between the solution of the tetrahedron equation and cluster algebras, enabling the construction of integrable systems with arbitrary symmetric Newton polygons and their embedding into double Bruhat cells.

Contribution

It introduces a new formalism connecting the tetrahedron equation with cluster algebra mutations and applies it to construct and classify integrable systems and conjugacy classes.

Findings

Connection between tetrahedron equation and cluster algebra mutations

Construction of integrable systems with arbitrary symmetric Newton polygons

Classification of conjugacy classes in double affine Weyl groups

Abstract

We notice a remarkable connection between 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 integrable system with spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into double Bruhat cell of a Poisson-Lie group, show how triangular decomposition can be used to extend our approach to general non-symmetric Newton polygons, and prove Lemma, which classifies conjugacy classes in double affine Weyl groups of $A$-type by Newton polygons.