On Discrete 3-Dimensional Equations Associated with the Local Yang-Baxter Relation

Abstract

The local Yang-Baxter equation (YBE), introduced by Maillet and Nijhoff, is a proper generalization to 3 dimensions of the zero curvature relation. Recently, Korepanov has constructed an infinite set of integrable 3-dimensional lattice models, and has related them to solutions to the local YBE. The simplest Korepanov's model is related to the star-triangle relation in the Ising model. In this paper the corresponding discrete equation is derived. In the continuous limit it leads to a differential 3d equation, which is symmetric with respect to all permutations of the three coordinates. A similar analysis of the star-triangle transformation in electric networks leads to the discrete bilinear equation of Miwa, associated with the BKP hierarchy. Some related operator solutions to the tetrahedron equation are also constructed.

10 pages, LaTeX, no figures

References in corpus (3)

• Current-Like Variables in Massive and Massless Integrable Models
• Algebraic integrable dynamical systems, 2+1-dimensional models in wholly discrete space-time, and inhomogeneous models in 2-dimensional statistical physics
• Affine Toda field theory as a 3-dimensional integrable system

Cited by in corpus (53)

• Zamolodchikov's Tetrahedron Equation and Hidden Structure of Quantum Groups
• Functional Tetrahedron Equation
• Direct linearising transform for three-dimensional discrete integrable systems: the lattice AKP, BKP and CKP equations
• Quantum geometry of 3-dimensional lattices
• Tetrahedron and 3D reflection equations from quantized algebra of functions
• Mathematical structure of three - dimensional (3D) Ising model
• Tetrahedron maps and symmetries of three dimensional integrable discrete equations
• Yang-Baxter maps associated to elliptic curves
• Degasperis-Procesi peakon dynamical system and finite Toda lattice of CKP type
• Discrete KP equation with self-consistent sources
• Non-commutative bi-rational maps satisfying Zamolodchikov equation, and Desargues lattices
• Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps
• Spinorial R-matrix
• The pentagon relation and incidence geometry
• A multidimensionally consistent version of Hirota's discrete KdV equation
• Operator approach to analytical evaluation of Feynman diagrams
• On exact solution of a classical 3D integrable model
• On Elliptic Lax Systems on the Lattice and a Compound Theorem for Hyperdeterminants
• From the Tetrahedron Equation to Universal R-Matrices
• Hyperdeterminants as integrable discrete systems
• Quantum Geometry of 3-Dimensional Lattices and Tetrahedron Equation
• Difference systems in bond and face variables and non-potential versions of discrete integrable systems
• T-systems, networks and dimers
• Spherical geometry and integrable systems
• Functional Relations on Anisotropic Potts Models: from Biggs Formula to the Tetrahedron Equation
• Discrete line complexes and integrable evolution of minors
• On a two dimensional system associated with the complex of the solutions of the Tetrahedron equation
• Solitons in a 3d integrable model
• Solutions of the functional tetrahedron equation connected with the local Yang -- Baxter equation for the ferro-electric
• Integrable lattices and their sublattices II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices
• Algebraic and differential-geometric constructions of set-theoretical solutions to the Zamolodchikov tetrahedron equation
• On reductions of the discrete Kadomtsev--Petviashvili-type equations
• On nonautonomous differential-difference AKP, BKP and CKP equations
• Duality for discrete integrable systems II
• Tetrahedron maps, Yang-Baxter maps, and partial linearisations
• Desargues maps and their reductions
• Lagrangian Grassmannians, CKP hierarchy and hyperdeterminantal relations
• Darboux transformations for linear operators on two dimensional regular lattices
• Discrete Darboux system with self-consistent sources and its symmetric reduction
• Pentagon Relations in Direct Sums and Grassmann Algebras
• Tau functions, infinite Grassmannians and lattice recurrences
• Multidimensional analogs of geometric s<-->t duality
• Eigenvector and eigenvalue problem for 3D bosonic model
• Integrable semi-discretisation of the Drinfel'd--Sokolov hierarchies
• Zamolodchikov Tetrahedral Equation and Higher Hamiltonians of 2d Quantum Integrable Systems
• Finite-dimensional analogs of string s <-> t duality and pentagon equation
• Tetrahedron equation and Schur functions
• The Kashaev Equation and Related Recurrences
• Ground states of Heisenberg evolution operator in discrete three-dimensional space-time and quantum discrete BKP equations
• Bäcklund transformations as integrable discretization. The geometric approach
• The free-fermionic $C_2^{(1)}$ loop model, double dimers and Kashaev's recurrence
• Penner coordinates for closed surfaces
• On the classification of multidimensionally consistent 3D maps