Affine Toda field theory as a 3-dimensional integrable system

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.

17 pages, LaTeX

References in corpus (5)

• Quantum Inverse Scattering Method and Correlation Functions
• Hirota equation as an example of integrable symplectic map
• 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
• Recent developments in affine Toda field theory

Cited by in corpus (7)

• On Discrete 3-Dimensional Equations Associated with the Local Yang-Baxter Relation
• Zamolodchikov's Tetrahedron Equation and Hidden Structure of Quantum Groups
• Quantum 2+1 evolution model
• Non-commutative bi-rational maps satisfying Zamolodchikov equation, and Desargues lattices
• From the Tetrahedron Equation to Universal R-Matrices
• Quantum Trilogy: Discrete Toda, Y-System and Chaos
• Integrable systems on singular symplectic manifolds: From local to global