Solutions of Discretized Affine Toda Field Equations for A_n⁽¹⁾, B_n⁽¹⁾, C_n⁽¹⁾, D_n⁽¹⁾, A_n⁽²⁾ and D_n+1⁽²⁾

Abstract

It is known that a family of transfer matrix functional equations, the T-system, can be compactly written in terms of the Cartan matrix of a simple Lie algebra.We formally replace this Cartan matrix of a simple Lie algebra with that of an affine Lie algebra, and then we obtain a system of functional equations different from the T-system. It may be viewed as an X_n^(a) type affine Toda field equation on discrete space time. We present, for A_n⁽¹⁾, B_n⁽¹⁾, C_n⁽¹⁾, D_n⁽¹⁾, A_n⁽²⁾ and D_n+1⁽²⁾, its solutions in terms of determinants or Pfaffians.