Tetrahedron Equation Research Articles

Solving the Yang-Baxter, tetrahedron and higher simplex equations using Clifford algebras

Bethe Ansatz was discovered in 1932. Half a century later its algebraic structure was unearthed: Yang-Baxter equation was discovered, as well as its multidimensional generalizations [tetrahedron equation and d-simplex equations]. Here we describe a universal method to solve these equations using Clifford algebras. The Yang-Baxter equation (d=2), Zamolodchikov's tetrahedron equation (d=3) and the Bazhanov-Stroganov equation (d=4) are special cases. Our solutions form a linear space. This helps us to include spectral parameters. Potential applications are discussed.

Tetrahedron duality

A certain two-dimensional supersymmetric gauge theory is argued to satisfy a relation that promotes the Zamolodchikov tetrahedron equation to an infrared duality between two quantum field theories. Solutions of the tetrahedron equation with continuous spin variables are obtained from partition functions of this theory and its variants.

Quantum Cluster Algebras and 3D Integrability: Tetrahedron and 3D Reflection Equations

Abstract We construct a new solution to the tetrahedron equation and the three-dimensional (3D) reflection equation by extending the quantum cluster algebra approach by Sun and Yagi concerning the former. We consider the Fock–Goncharov quivers associated with the longest elements of the Weyl groups of type $A$ and $C$, and investigate the cluster transformations corresponding to changing a reduced expression into a “most distant” one. By devising a new realization of the quantum $y$-variables in terms of $q$-Weyl algebra, the solutions are extracted as the operators whose adjoint actions yield the cluster transformations of the quantum $y$-variables. Explicit formulas of their matrix elements are also derived for some typical representations.

Electric network and Hirota type 4-simplex maps

Bazhanov-Stroganov (4-simplex) maps are set-theoretical solutions to the 4-simplex equation, namely the fourth member of the family of n-simplex equations, which are fundamental equations of mathematical physics. In this paper, we develop a method for constructing Bazhanov-Stroganov maps as extensions of tetrahedron maps which are set-theoretical solutions to the Zamolodchikov tetrahedron (3-simplex) equation. We employ this method to construct birarional Bazhanov-Stroganov maps which boil down to the famous electric network and Hirota tetrahedron maps at a certain limit.

On the solutions of the local Zamolodchikov tetrahedron equation

We study the solutions of the local Zamolodhcikov tetrahedron equation in the form of correspondences derived by 3×3 matrices with free variables. We present all the associated generators of 4-simplex maps satisfying the local tetrahedron equation. Moreover, we demonstrate that, from some of our solutions, we can recover the 4-simplex extensions of Kashaev–Korepanov–Sergeev and Hirota type tetrahedron maps. Finally, we construct several novel 4-simplex maps.

Tetrahedron equation and quantum cluster algebras

We develop the quantum cluster algebra approach recently introduced by Sun and Yagi to investigate the tetrahedron equation, a three-dimensional generalization of the Yang-Baxter equation. In the case of square quiver, we devise a new realization of quantum Y-variables in terms q-Weyl algebras and obtain a solution that possesses three spectral parameters. It is expressed in various forms, comprising four products of quantum dilogarithms depending on the signs in decomposing the quantum mutations into the automorphism part and the monomial part. For a specific choice of them, our formula precisely reproduces Sergeev’s R matrix, which corresponds to a vertex formulation of the Zamolodchikov-Bazhanov-Baxter model when q is specialized to a root of unity.

Local Yang–Baxter correspondences and set-theoretical solutions to the Zamolodchikov tetrahedron equation

We study tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation, and their matrix Lax representations defined by the local Yang–Baxter equation. Sergeev (1998 Lett. Math. Phys. 45 113–9) presented classification results on three-dimensional tetrahedron maps obtained from the local Yang–Baxter equation for a certain class of matrix-functions in the situation when the equation possesses a unique solution which determines a tetrahedron map. In this paper, using correspondences arising from the local Yang–Baxter equation for some simple matrix-functions, we show that there are (non-unique) solutions to the local Yang–Baxter equation which define tetrahedron maps that do not belong to the Sergeev list; this paves the way for a new, wider classification of tetrahedron maps. We present invariants for the derived tetrahedron maps and prove Liouville integrability for some of them. Furthermore, using the approach of solving correspondences arising from the local Yang–Baxter equation, we obtain several new birational tetrahedron maps with Lax representations and invariants, including maps on arbitrary groups, a nine-dimensional map associated with a Darboux transformation for the derivative nonlinear Schrödinger equation, and a nine-dimensional generalisation of the three-dimensional Hirota map.

New Solutions to the Tetrahedron Equation Associated with Quantized Six-Vertex Models

We present a family of new solutions to the tetrahedron equation of the form RLLL=LLLR, where L operator may be regarded as a quantized six-vertex model whose Boltzmann weights are specific representations of the q-oscillator or q-Weyl algebras. When the three L’s are associated with the q-oscillator algebra, R coincides with the known intertwiner of the quantized coordinate ring A_q(sl_3). On the other hand, L’s based on the q-Weyl algebra lead to new R’s whose elements are either factorized or expressed as a terminating q-hypergeometric type series.

Birational solutions to the set-theoretical 4-simplex equation

The 4-simplex equation is a higher-dimensional analogue of Zamolodchikov’s tetrahedron equation and the Yang–Baxter equation which are two of the most fundamental equations of mathematical physics. In this paper, we introduce a method for constructing 4-simplex maps, namely solutions to the set-theoretical 4-simplex equation, using Lax matrix refactorisation problems. Employing this method, we construct 4-simplex maps which at a certain limit give tetrahedron maps classified by Kashaev, Korepanov and Sergeev. Moreover, we construct a Kadomtsev–Petviashvili type of 4-simplex map. Finally, we introduce a method for constructing 4-simplex maps which can be restricted on level sets to parametric 4-simplex maps using Darboux transformations of integrable PDEs. We construct a nonlinear Schrödinger type parametric 4-simplex map which is the first parametric 4-simplex map in the literature.

Integrable 3D lattice model in M-theory

It is argued that the supersymmetric index of a certain system of branes in M-theory is equal to the partition function of an integrable three-dimensional lattice model. The local Boltzmann weights of the lattice model satisfy a generalization of Zamolodchikov’s tetrahedron equation. In a special case the model is described by a solution of the tetrahedron equation discovered by Kapranov and Voevodsky and by Bazhanov and Sergeev.

Cluster transformations, the tetrahedron equation, and three-dimensional gauge theories

Cluster transformations, the tetrahedron equation, and three-dimensional gauge theories

Algebraic and differential-geometric constructions of set-theoretical solutions to the Zamolodchikov tetrahedron equation

We present several algebraic and differential-geometric constructions of tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation. In particular, we obtain a family of new (nonlinear) polynomial tetrahedron maps on the space of square matrices of arbitrary size, using a matrix refactorisation equation, which does not coincide with the standard local Yang–Baxter equation. Liouville integrability is established for some of these maps. Also, we show how to derive linear tetrahedron maps as linear approximations of nonlinear ones, using Lax representations and the differentials of nonlinear tetrahedron maps on manifolds. We apply this construction to two nonlinear maps: a tetrahedron map obtained in Dimakis and Müller-Hoissen (2019 Lett. Math. Phys. 109 799–827) in a study of soliton solutions of vector Kadomtsev–Petviashvili equations and a tetrahedron map obtained in Konstantinou-Rizos (2020 Nucl. Phys. B 960 115207) in a study of a matrix trifactorisation problem related to a Darboux matrix associated with a Lax operator for the nonlinear Schrödinger equation. We derive parametric families of new linear tetrahedron maps (with nonlinear dependence on parameters), which are linear approximations for these nonlinear ones. Furthermore, we present (nonlinear) matrix generalisations of a tetrahedron map from Sergeev’s classification Sergeev (1998 Lett. Math. Phys. 45 113–9). These matrix generalisations can be regarded as tetrahedron maps in noncommutative variables. Besides, several tetrahedron maps on arbitrary groups are constructed.

Set-theoretical solutions of the Zamolodchikov tetrahedron equation on associative rings and Liouville integrability

This paper is devoted to tetrahedron maps, which are set-theoretical solutions of the Zamolodchikov tetrahedron equation. We construct a family of tetrahedron maps on associative rings. The obtained maps are new to our knowledge. We show that matrix tetrahedron maps derived previously are a particular case of our construction. This provides an algebraic explanation of the fact that the matrix maps satisfy the tetrahedron equation. Also, Liouville integrability is established for some of the constructed maps.

Noncommutative solutions to Zamolodchikov’s tetrahedron equation and matrix six-factorisation problems

It is known that the local Yang–Baxter equation is a generator of potential solutions to Zamolodchikov’s tetrahedron equation. In this paper, we show under which additional conditions the solutions to the local Yang–Baxter equation are tetrahedron maps, namely solutions to the set-theoretical tetrahedron equation. This is exceptionally useful when one wants to prove that noncommutative maps satisfy the Zamolodchikov’s tetrahedron equation. We construct new noncommutative maps and we prove that they possess the tetrahedron property. Moreover, by employing Darboux transformations with noncommutative variables, we derive noncommutative tetrahedron maps. In particular, we derive a noncommutative nonlinear Schrödinger type of tetrahedron map which can be restricted to a noncommutative version of Sergeev’s map on invariant leaves. We prove that these maps are tetrahedron maps.

Set-theoretical solutions of the Zamolodchikov tetrahedron equation on associative rings and Liouville integrability

Изучаются отображения тетраэдров, которые являются теоретико-множественными решениями уравнения тетраэдров Замолодчикова. Построено семейство отображений тетраэдров на ассоциативных кольцах. Полученные отображения являются, насколько нам известно, новыми. Показано, что полученные ранее матричные отображения тетраэдров являются частным случаем нашей конструкции. Это дает алгебраическое объяснение того факта, что эти матричные отображения удовлетворяют уравнению тетраэдров. Также для некоторых из построенных отображений установлена интегрируемость по Лиувиллю.

Noncommutative Solutions to Zamolodchikov's Tetrahedron Equation and Matrix Six-Factorisation Problems

Noncommutative Solutions to Zamolodchikov's Tetrahedron Equation and Matrix Six-Factorisation Problems

Tetrahedron maps, Yang–Baxter maps, and partial linearisations

We study tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation, and Yang–Baxter maps, which are set-theoretical solutions to the quantum Yang–Baxter equation. In particular, we clarify the structure of the nonlinear algebraic relations which define linear (parametric) tetrahedron maps (with nonlinear dependence on parameters), and we present several transformations which allow one to obtain new such maps from known ones. Furthermore, we prove that the differential of a (nonlinear) tetrahedron map on a manifold is a tetrahedron map as well. Similar results on the differentials of Yang–Baxter and entwining Yang–Baxter maps are also presented. Using the obtained general results, we construct new examples of (parametric) Yang–Baxter and tetrahedron maps. The considered examples include maps associated with integrable systems and matrix groups. In particular, we obtain a parametric family of new linear tetrahedron maps, which are linear approximations for the nonlinear tetrahedron map constructed by Dimakis and Müller-Hoissen (2019 Lett. Math. Phys. 109 799–827) in a study of soliton solutions of vector Kadomtsev–Petviashvili equations. Also, we present invariants for this nonlinear tetrahedron map.

Kirillov–Reshetikhin Modules of Generalized Quantum Groups of Type $A$

The generalized quantum group of type A is an affine analogue of the quantum group associated to a general linear Lie superalgebra, which appears in the study of solutions to the tetrahedron equation or the three-dimensional Yang–Baxter equation. In this paper we develop the crystal base theory for finite-dimensional representations of generalized quantum groups of type A . As the main result, we construct Kirillov–Reshetikhin modules, that is, a family of irreducible modules which have crystal bases. We also give an explicit combinatorial description of the crystal structure of Kirillov–Reshetikhin modules, the combinatorial R matrix and the energy function on their tensor products.

Tetrahedron equation: algebra, topology, and integrability

The Zamolodchikov tetrahedron equation inherits almost all the richness of structures and topics in which the Yang–Baxter equation is involved. At the same time, this transition symbolizes the growth of the order of the problem, the step from the Yang–Baxter equation to the local Yang–Baxter equation, from the Lie algebra to the 2-Lie algebra, from ordinary knots in to 2-knots in . These transitions are followed in several examples, and there are also discussions of the manifestation of the tetrahedron equation in the long-standing question of integrability of the three-dimensional Ising model and a related model of neural network theory: the Hopfield model on a two-dimensional lattice. Bibliography: 82 titles.

Boundary from Bulk Integrability in Three Dimensions: 3D Reflection Maps from Tetrahedron Maps

We established a method for obtaining set-theoretical solutions to the 3D reflection equation by using known ones to the Zamolodchikov tetrahedron equation, where the former equation was proposed by Isaev and Kulish as a boundary analog of the latter. By applying our method to Sergeev's electrical solution and a two-component solution associated with the discrete modified KP equation, we obtain new solutions to the 3D reflection equation. Our approach is closely related to a relation between the transition maps of Lusztig's parametrizations of the totally positive part of $SL_3$ and $SO_5$, which is obtained via folding the Dynkin diagram of $A_3$ into one of $B_2$.