A2 Toda theory in reduced WZNW framework and the representations of the   W algebra
  Using the reduced WZNW formulation we analyse the classical $W$ orbit content of the space of classical solutions of the $A_2$ Toda theory. We define the quantized Toda field as a periodic primary field of the $W$ algebra satisfying the quantized equations of motion. We show that this local operator can be constructed consistently only in a Hilbert space consisting of the representations corresponding to the minimal models of the $W$ algebra. 
Z. Bajnok, L.Palla, G. Takacs