Zamolodchikov's Tetrahedron Equation and Hidden Structure of Quantum Groups

TL;DR

This paper introduces a new solution to the tetrahedron equation that reveals a hidden three-dimensional structure in quantum groups, leading to new 2D integrable models and a rank-size duality in their Bethe Ansatz solutions.

Contribution

The paper constructs a novel solution to the tetrahedron equation linking 3D integrable models with quantum affine algebras, unveiling a hidden 3D structure in known 2D models.

Findings

New solution of the tetrahedron equation for all quantum affine algebras

Revealed hidden 3D structure in 2D integrable models

Established a rank-size duality in Bethe Ansatz solutions

Abstract

The tetrahedron equation is a three-dimensional generalization of the Yang-Baxter equation. Its solutions define integrable three-dimensional lattice models of statistical mechanics and quantum field theory. Their integrability is not related to the size of the lattice, therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubic lattices. Obviously, any such three-dimensional model can be viewed as a two-dimensional integrable model on a square lattice, where the additional third dimension is treated as an internal degree of freedom. Therefore every solution of the tetrahedron equation provides an infinite sequence of integrable 2d models differing by the size of this "hidden third dimension". In this paper we construct a new solution of the tetrahedron equation, which provides in this way the two-dimensional solvable models related to finite-dimensional highest weight representations for all quantum affine algebra $U_q(\hat{sl}(n))$, where the rank $n$ coincides with the size of the hidden dimension. These models are related with an anisotropic deformation of the $sl(n)$-invariant Heisenberg magnets. They were extensively studied for a long time, but the hidden 3d structure was hitherto unknown. Our results lead to a remarkable exact "rank-size" duality relation for the nested Bethe Ansatz solution for these models. Note also, that the above solution of the tetrahedron equation arises in the quantization of the "resonant three-wave scattering" model, which is a well-known integrable classical system in 2+1 dimensions.