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 tau-variables is found. The thermodynamic Bethe ansatz of the affine Toda system is studied in the limit L, N --> infinity. It is shown that the free energy of the systems grows proportionally to the volume.