Remarks on $A_2$ Toda theory

We study the Toda field theory with finite Lie algebras using an extension of the Goulian–Li technique. In this way, we show that, after integrating over the zero mode in the correlation functions of the exponential fields, the resulting correlation function resembles that of a free theory. Furthermore, it is shown that for some ratios of the charges of the exponential fields the four-point correlation functions which contain a degenerate field satisfy the Riemann ordinary differential equation. Using this fact and the crossing symmetry, we derive a set of functional equations for the structure constants of the $A_2$ Toda field theory.