Birational solutions to the set-theoretical 4-simplex equation

TL;DR

This paper introduces a novel method for constructing solutions to the set-theoretical 4-simplex equation using Lax matrix refactorisation, leading to new classes of 4-simplex maps, including parametric ones related to integrable PDEs.

Contribution

It presents a new construction method for 4-simplex maps via Lax matrix refactorisation, including the first parametric 4-simplex map derived from Darboux transformations.

Findings

Constructed 4-simplex maps from Lax matrix refactorisation.

Connected certain 4-simplex maps to known tetrahedron maps.

Developed a nonlinear Schrödinger type parametric 4-simplex map.

Abstract

The 4-simplex equation is a higher-dimensional analogue of Zamolodchikov's tetrahedron equation and the Yang--Baxter equation which are two of the most fundamental equations of mathematical physics. In this paper, we introduce a method for constructing 4-simplex maps, namely solutions to the set-theoretical 4-simplex equation, using Lax matrix refactorisation problems. Employing this method, we construct 4-simplex maps which at a certain limit give tetrahedron maps classified by Kashaev, Korepanov and Sergeev. Moreover, we construct a Kadomtsev--Petviashvili type of 4-simplex map. Finally, we introduce a method for constructing 4-simplex maps which can be restricted on level sets to parametric 4-simplex maps using Darboux transformations of integrable PDEs. We construct a nonlinear Schr\"odinger type parametric 4-simplex map which is the first parametric 4-simplex map in the literature.