Clifford Solver for the Tetrahedron Equation and its Variants

TL;DR

This paper demonstrates that Clifford algebras can solve various forms of the tetrahedron equation, advancing understanding of three-dimensional integrability and providing a scheme for solving higher simplex equations.

Contribution

It introduces Clifford algebra solutions for multiple tetrahedron equation variants and proposes a canonical scheme for higher simplex equations.

Findings

Clifford algebras solve constant and spectral parameter tetrahedron equations.

Solutions extend to variants of the tetrahedron equation.

A scheme for solving higher simplex equations is presented.

Abstract

The different forms of the tetrahedron equation appear when all possible ways to label the scattering process of infinitely long straight lines are considered in three dimensional spacetime. This is expected to lead to three dimensional integrability, analogous to the Yang-Baxter equation. Among the three possibilities, we consider two of them and their variants. We show that Clifford algebras solve both the constant and the spectral parameter dependent versions of all of them. We also present a scheme for canonically solving higher simplex equations using tetrahedron solutions.