Tetrahedron Equation Research Articles

STAR-SQUARE AND TETRAHEDRON EQUATIONS IN THE BAXTER-BAZHANOV MODEL

Spatially symmetrical Baxter-Bazhanov model’s Boltzmann weights are proved to satisfy the tetrahedron equation. The latter is a consequence of the “inversion” and “star-square” relations for the Boltzmann weight elementary building blocks.

SPATIAL SYMMETRY, LOCAL INTEGRABILITY AND TETRAHEDRON EQUATIONS IN THE BAXTER-BAZHANOV MODEL

We show that the Baxter-Bazhanov model is invariant under the action of the cube’s symmetry group. The three-dimensional star-star relations, proposed by Baxter and Bazhanov as local integrability conditions, correspond to a particular transformation from this group. Invariant Boltzmann weights, parametrized in terms of the Zamolodchikov angle variables, apparently satisfy the tetrahedron equations.

Some constant solutions to Zamolodchikov's tetrahedron equations

In this letter we present constant solutions to the tetrahedron equations proposed by Zamolodchikov. In general, from a given solution of the Yang-Baxter equation there are two ways to construct solutions to the tetrahedron equation. There are also other kinds of solutions. We present some two-dimensional solutions that were obtained by directly solving the equations using either an upper triangular or Zamolodchikov's ansatz.

ON THE SYMMETRIES OF INTEGRABILITY

We show that the Yang-Baxter equations for two-dimensional models admit as a group of symmetry the infinite discrete group [Formula: see text]. The existence of this symmetry explains the presence of a spectral parameter in the solutions of the equations. We show that similarly, for three-dimensional vertex models and the associated tetrahedron equations, there also exists an infinite discrete group of symmetry. Although generalizing naturally the previous one, it is a much bigger hyperbolic Coxeter group. We indicate how this symmetry can help to resolve the Yang-Baxter equations and their higher-dimensional generalizations and initiate the study of three-dimensional vertex models. These symmetries are naturally represented as birational projective transformations. They may preserve non-trivial algebraic varieties, and lead to proper parametrizations of the models, be they integrable or not. We mention the relation existing between spin models and the Bose-Messner algebras of algebraic combinatorics. Our results also yield the generalization of the condition qn=1 so often mentioned in the theory of quantum groups, when no q parameter is available.

Rational mappings, arborescent iterations, and the symmetries of integrability.

We show that the Yang-Baxter equations for two dimensional models admit as a group of symmetry the infinite discrete group $A_2^{(1)}$. The existence of this symmetry explains the presence of a spectral parameter in the solutions of the equations. We show that similarly, for three-dimensional vertex models and the associated tetrahedron equations, there also exists an infinite discrete group of symmetry. Although generalizing naturally the previous one, it is a much bigger hyperbolic Coxeter group. We indicate how this symmetry can help to resolve the Yang-Baxter equations and their higher-dimensional generalizations and initiate the study of three-dimensional vertex models. These symmetries are naturally represented as birational projective transformations. They may preserve non trivial algebraic varieties, and lead to proper parametrizations of the models, be they integrable or not. We mention the relation existing between spin models and the Bose-Messner algebras of algebraic combinatorics. Our results also yield the generalization of the condition $q^n=1$ so often mentioned in the theory of quantum groups, when no $q$ parameter is available.

Simplex equations and their solutions

We investigate ann-simplex generalization of the classical and quantum Yang-Baxter equation. For the case ofsl(2) we find the most general solution of the classicaln-simplex equation for alln. These classical solutions can be quantized (in the sense of quantum group theory) forn=2,3 and we exhibit a quantum solution to the tetrahedron equations (n=3). The classical nondegenerate solutions cannot be quantized forn=4.

Infinite discrete symmetry group for the Yang-Baxter equations. Vertex models

Abstract We show that the Yang-Baxter equations for two-dimensional vertex models admit as a group of symmetry the infinite discrete group A 2 (1) . The existence of this symmetry explains the presence of a spectral parameter in solutions of the equations. We show that similarly, for three-dimensional vertex models and the associated tetrahedron equations, there also exists an infinite discrete group of symmetries. Although generalizing very naturally the previous one, this is a much bigger hyperbolic Coxeter group. We indicate how this symmetry should be used to resolve the Yang-Baxter equations and their higher-dimensional generalizations and initiate the study of a family of three-dimensional vertex models.

TESTING THE BETHE ANSATZ FOR STRAIGHT STRINGS IN (2 + 1) DIMENSIONS

The general quantum theory of nonrelativistic, infinitely long straight strings of one flavor interacting via a three-body infinitesimal range potential is considered in (2 + 1) dimensions. It is proved that the tetrahedron equation of Zamolodchikov is satisfied only when the potential is either zero or infinitely repulsive. Therefore the Bethe ansatz is valid only for these special cases.

TETRAHEDRAL ZAMOLODCHIKOV ALGEBRA AND THE TWO-LAYER FLAT MODEL IN STATISTICAL MECHANICS

The tetrahedral Zamolodchikov algebra arises in the theory of the tetrahedron equations. From this algebra, the solutions of triangle equations corresponding to a model on the two neighboring layers of the 3-dimensional cubic lattice are obtained.

The tetrahedron equation and the four-simplex equation

The tetrahedron equation and the four-simplex equation are multidimensional generalizations of the Yang-Baxter or triangle equations. We discuss common features of these members of the family of “simplex equations”. Zamolodchikov's solution of the tetrahedron equation is rewritten in an algebraic form and a generalization of it to the four-simplex case is proposed. Relevance of the simplex equation for the understanding of multidimensional integrability is briefly discussed.

Hunting for d > 2 quantum integrable systems on the computer

A simple numerical method is described that discriminates between d = 3 (2) systems that can satisfy the tetrahedron (triangle) equations of integrability and those that cannot. As an application, it is argued that a certain d = 3 model of the Baxter type is not integrable.

Free fermions on a three-dimensional lattice and tetrahedron equations

Abstract A method to construct a commuting transfer matrix has been proposed for three-dimensional fermion field theory. The method is based on the use of “tetrahedron equations”. For the case of free fermions, the commuting transfer matrix structure has been studied completely and some solution has been obtained for the tetrahedron equations.

On Zamolodchikov's solution of the tetrahedron equations

The tetrahedron equations arise in field theory as the condition for theS-matrix in 2+1-dimensions to be factorizable, and in statistical mechanics as the condition that the transfer matrices of three-dimensional models commute. Zamolodchikov has proposed what appear (from numerical evidence and special cases) to be non-trivial particular solutions of these equations, but has not fully verified them. Here it is proved that they are indeed solutions.

Tetrahedron equations and the relativisticS-matrix of straight-strings in 2+1-Dimensions

The quantumS-matrix theory of straight-strings (infinite one-dimensional objects like straight domain walls) in 2+1-dimensions is considered. TheS-matrix is supposed to be “purely elastic” and factorized. The tetrahedron equations (which are the factorization conditions) are investigated for the special “two-colour” model. The relativistic three-stringS-matrix, which apparently satisfies this tetrahedron equation, is proposed.

Solutions to the Vector Tetrahedron Equation

A set of nine closed-form solutions are presented to the single, three-dimensional vector tetrahedron equation, sum of vectors equals zero. The set represents all possible combinations of unknown spherical coordinates among the vectors, assuming the coordinates are functionally independent. Optimum use is made of symmetry. The solutions are interpretable and can be evaluated reliably by digital computer in milliseconds. They have been successfully applied to position determination of many three-dimensional mechanisms.