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

TL;DR

This paper explores discrete integrable field theories in 2+1 dimensions using circular net equations, classifies physical regimes via geometric criteria, and links spectral parameters to various physical models and phenomena.

Contribution

It introduces a classification of regimes for discrete integrable field theories based on geometric criteria and analyzes the role of spectral parameters in physical and statistical models.

Findings

Four distinct physical regimes classified by geometric triangles.

Existence of ground states and solitons due to spectral parameters.

Connections between spectral parameters and statistical mechanics or field theories.

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.