Exact Operator Solution of A2-Toda Field Theory

Quantum A2-Toda field theory in two dimensions is investigated based on the method of quantizing canonical free field. Toda exponential operators associated with the fundamental weights are constructed to the fourth order in the cosmological constant. This leads us to a conjecture for the exact operator solution. It is well-known that Toda field theories in two dimensions admit exact classical solutions [1]. Recently, they have attracted renewed interests as conformal field theories incorporating W symmetry [2, 3, 4]. The simplest Toda theory is the Liouville theory, for which extensive canonical approaches have been developed so far [5, 6, 7]. This includes the exact operator solution [8, 9, 10, 11]. Though similar development have been achieved also for Toda theories [12, 13, 3], complete operator solution seems to be still lacking. In this note we investigate exact operator solution of A2-Toda field theory by the method of quantizing canonical free field developed for Liouville theory [5, 6, 7]. We shall closely follow the arguments due to Otto and Weigt [8] in constructing the operator solution for Liouville theory. It should be noted, however, that the extension of their method to Toda theories is not so obvious since the latter contain more than one screening charges which are not mutually commuting in general. Furthermore, Toda exponential operators can be parametrized by an arbitrary vector in the root space. These brings about an operator ordering problem in expanding the Toda exponentials in terms of the screening charges. Such complications may be resolved by considering special operators associated with the fundamental weights. We find that only commuting screening charges appear in the expansions of these operators. Their quantum expressions can be determined order by order in the cosmological constant from the requirement of locality [6, 8]. General exponential operators as well as the Toda field itself can be obtained from them. Conformality and locality alone determine the exact solution up to a constant related to the arbitrariness of the cosmological constant. This ambiguity is resolved by imposing the field equation. The canonicity of the transformation from the Toda field to the free field at the quantum level can be established directly by examining the canonical commutation relations. Let us consider A2-Toda field theory described by the classical action