Solving the tetrahedron equation by Teichm\"uller TQFT

TL;DR

This paper introduces a method to build 3D lattice models with integrable properties using Teichmüller TQFT, connecting geometric structures with quantum field theory to solve the tetrahedron equation.

Contribution

It presents a novel approach to construct integrable 3D lattice models via state integral models and Teichmüller TQFT, providing explicit solutions to the tetrahedron equation.

Findings

Boltzmann weights satisfy a variant of the tetrahedron equation

Constructs integrable 3D lattice models from geometric data

Provides explicit example using Teichmüller TQFT

Abstract

We propose an approach to construct three-dimensional lattice models using line defects in state integral models on shaped triangulations of 3-manifolds. The Boltzmann weights for these models satisfy a variant of the tetrahedron equation, which implies integrability under suitable assumptions on R-matrices and transfer matrices. As an explicit example, we present a solution produced by Teichm\"uller TQFT.