Zamolodchikov's tetrahedron equation and hidden structure of quantum groups

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 finitedimensional highest weight representations for all quantum affine algebra U q ( 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.

Multispecies totally asymmetric zero range process: II. Hat relation and tetrahedron equation

“…Here and in what follows, the components of the tensor product will always be ordered so that they correspond, from left to right, to the vertices (if exist) at (1, 1), (2, 1), (1, 2), (3, 1), (2,2), (1, 3), . .…”

“…By the construction it satisfies the constant tetrahedron equation. The R was also given in [2,3] in a different gauge from a quantum geometry consideration. The two were identified in [16, eq.(2.29)].…”

“…It was also given in [2] and the two were identified in [16]. It defines a vertex model on a cubic lattice whose edges are assigned with spin variables in Z ≥0 and vertices with polynomial Boltzmann weights in q.…”

Tetrahedron Equation and Quantum R Matrices for Spin Representations of $${B^{(1)}_n}$$ B n ( 1 ) , $${D^{(1)}_n}$$ D n ( 1 ) and $${D^{(2)}_{n+1}}$$ D n + 1 ( 2 )

“…This phenomenon is known as dimension-rank transmutation. It has been implemented earlier for a certain 3d L operator by taking the trace which corresponds to the periodic boundary condition in the hidden direction [4,5,6]. The resulting solutions of the Yang-Baxter equation have been identified with the quantum R matrices for a class of finite dimensional representations of U q ( sl n ).…”

“…We start from the solution of the tetrahedron equation consisting of q-oscillator 3d R matrix and fermionic 3d L operators, which are the same as [4]. We construct special boundary states in a bosonic Fock space (the hidden direction) and show that they are eigenvectors of the 3d R-matrix, which is the key to make our reduction scheme work.…”