Affine Toda field theory as a 3-dimensional integrable system

TL;DR

This paper explores the affine Toda field theory as a 2+1-dimensional integrable system with a discrete third dimension, introducing a natural discretization and analyzing its spectral properties through thermodynamic Bethe ansatz.

Contribution

It introduces a natural discretization of affine Toda theory in 2+1 dimensions and constructs the discrete evolution operator explicitly.

Findings

Existence of a permutation-invariant discretization of affine Toda equations

Explicit construction of the discrete evolution operator

Conjectures on the spectrum structure in the thermodynamic limit

Abstract

The affine Toda field theory is studied as a 2+1-dimensional system. The third dimension appears as the discrete space dimension, corresponding to the simple roots in the $A_N$ affine root system, enumerated according to the cyclic order on the $A_N$ affine Dynkin diagram. We show that there exists a natural discretization of the affine Toda theory, where the equations of motion are invariant with respect to permutations of all discrete coordinates. The discrete evolution operator is constructed explicitly. The thermodynamic Bethe ansatz of the affine Toda system is studied in the limit $L,N\to\infty$. Some conjectures about the structure of the spectrum of the corresponding discrete models are stated.