Affine Toda Field Theory as a 3-Dimensional Integrable System

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. The quantum analog of the τ-variables is found. The thermodynamic Bethe ansatz of the affine Toda system is studied in the limit L,N→∞. It is shown that the free energy of the systems grows proportionally to the volume.