Solving the Yang-Baxter, tetrahedron and higher simplex equations using Clifford algebras

TL;DR

This paper introduces a universal method leveraging Clifford algebras to solve the Yang-Baxter, tetrahedron, and higher simplex equations, unifying their solutions and enabling spectral parameter inclusion.

Contribution

It presents a novel, algebraic approach using Clifford algebras to solve multidimensional integrability equations, extending solutions to include spectral parameters.

Findings

Solutions form a linear space

Method applies to Yang-Baxter, tetrahedron, and 4-simplex equations

Potential applications in integrable models

Abstract

Bethe Ansatz was discoverd in 1932. Half a century later its algebraic structure was unearthed: Yang-Baxter equation was discovered, as well as its multidimensional generalizations [tetrahedron equation and $d$-simplex equations]. Here we describe a universal method to solve these equations using Clifford algebras. The Yang-Baxter equation ($d=2$), Zamalodchikov's tetrahedron equation ($d=3$) and the Bazhanov-Stroganov equation ($d=4$) are special cases. Our solutions form a linear space. This helps us to include spectral parameters. Potential applications are discussed.