Functional Tetrahedron Equation

Abstract

We describe a scheme of constructing classical integrable models in 2+1-dimensional discrete space-time, based on the functional tetrahedron equation - equation that makes manifest the symmetries of a model in local form. We construct a very general "block-matrix model" together with its algebro-geometric solutions, study its various particular cases, and also present a remarkably simple scheme of quantization for one of those cases.

LaTeX, 16 pages

References in corpus (2)

• Solutions of the functional tetrahedron equation connected with the local Yang -- Baxter equation for the ferro-electric
• Integrability in 3+1 Dimensions: Relaxing a Tetrahedron Relation

Cited by in corpus (42)

• 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
• Non-commutative bi-rational maps satisfying Zamolodchikov equation, and Desargues lattices
• Tetrahedron Equation and Quantum R Matrices for Spin Representations of B^{(1)}_n, D^{(1)}_n and D^{(2)}_{n+1}
• Invariants in Separated Variables: Yang-Baxter, Entwining and Transfer Maps
• Electrical varieties as vertex integrable statistical models
• Super-tetrahedra and super-algebras
• Matrix KP: tropical limit and Yang-Baxter maps
• Cohomologies of $n$-simplex relations
• Entwining Yang-Baxter maps related to NLS type equations
• Yang--Baxter maps, Darboux transformations, and linear approximations of refactorisation problems
• Quantum Geometry of 3-Dimensional Lattices and Tetrahedron Equation
• From the Tetrahedron Equation to Universal R-Matrices
• Difference systems in bond and face variables and non-potential versions of discrete integrable systems
• Solving the Yang-Baxter, tetrahedron and higher simplex equations using Clifford algebras
• Quantization of three-wave equations
• Local Yang--Baxter correspondences and set-theoretical solutions to the Zamolodchikov tetrahedron equation
• Algebraic and differential-geometric constructions of set-theoretical solutions to the Zamolodchikov tetrahedron equation
• Solution of tetrahedron equation and cluster algebras
• Birational solutions to the set-theoretical 4-simplex equation
• Tetrahedron maps, Yang-Baxter maps, and partial linearisations
• Non-commutative double-sided continued fractions
• Pentagon Relations in Direct Sums and Grassmann Algebras
• Boundary from bulk integrability in three dimensions: 3D reflection maps from tetrahedron maps
• The Positive orthogonal Grassmannian and loop amplitudes of ABJM
• Multidimensional analogs of geometric s<-->t duality
• Eigenvector and eigenvalue problem for 3D bosonic model
• Systems of difference equations on a vector valued function that admit 3D space of scalar potentials
• Discrete nonlinear hyperbolic equations. Classification of integrable cases
• Zamolodchikov Tetrahedral Equation and Higher Hamiltonians of 2d Quantum Integrable Systems
• Singularization of knots and closed braids
• Quantum one-cocycles for knots
• Finite-dimensional analogs of string s <-> t duality and pentagon equation
• Geometric torsions and invariants of manifolds with triangulated boundary
• Set-theoretical solutions to the Zamolodchikov tetrahedron equation on associative rings and Liouville integrability
• Tetrahedron equation and Schur functions
• Noncommutative solutions to Zamolodchikov's tetrahedron equation and matrix six-factorisation problems
• The tetrahedral analog of Veneziano amplitude
• Geometry of quadrilateral nets: second Hamiltonian form
• Ground states of Heisenberg evolution operator in discrete three-dimensional space-time and quantum discrete BKP equations
• Penner coordinates for closed surfaces