Classical integrable field theories in discrete 2+1 dimensional space-time

Abstract

We study "circular net" (discrete orthogonal net) equations for angular data generalized by external spectral parameters. A criterion defining physical regimes of these Hamiltonian equations is the reality of Lagrangian density. There are four distinct regimes for fields and spectral parameters classified by four types of spherical or hyperbolic triangles. Non-zero external spectral parameters provide the existence of field-theoretical ground states and soliton excitations. Spectral parameters of a spherical triangle correspond to a statistical mechanics; spectral parameters of hyperbolic triangles correspond to three different field theories with massless anisotropic dispersion relations.

22 pages

References in corpus (9)

• Discrete Heisenberg-Weyl Group and Modular Group
• Multidimensional Quadrilateral Lattices are Integrable
• Zamolodchikov's Tetrahedron Equation and Hidden Structure of Quantum Groups
• Discrete differential geometry. Consistency as integrability
• Algebraic integrable dynamical systems, 2+1-dimensional models in wholly discrete space-time, and inhomogeneous models in 2-dimensional statistical physics
• Quantum geometry of 3-dimensional lattices
• Super-tetrahedra and super-algebras
• Tetrahedron equation and the algebraic geometry
• Quantization of three-wave equations

Cited by in corpus (4)

• An integrable 3D lattice model with positive Boltzmann weights
• Integrable Structure of Multispecies Zero Range Process
• Combinatorial Yang-Baxter maps arising from tetrahedron equation
• Ground states of Heisenberg evolution operator in discrete three-dimensional space-time and quantum discrete BKP equations