Clifford Solver for the Tetrahedron Equation and its Variants

TL;DR

The paper targets three-dimensional integrability by studying the tetrahedron equation and its variants, arising from labeling scattering processes in 3D spacetime. It introduces a universal Clifford-algebra-based method to construct solutions that satisfy both constant and spectral-parameter versions of vertex and edge forms, including the Frenkel–Moore and related $RLLL$/$MMLL$ equations. A canonical scheme is developed to build higher simplex equations by decomposing simplex operators into tetrahedron factors, with explicit construction for the 4-simplex and generalization via an index-splitting rule. The results provide a constructive toolkit for 3D solvable models and higher-dimensional algebras, with potential applications in statistical physics through positive Boltzmann weights and connections to other algebraic techniques like Majorana fermions and partition algebras.

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.