Integrable Hierarchies and Wakimoto Modules

In our papers [20, 21] we proposed a new approach to integrable hierarchies of soliton equations and their quantum deformations. We have applied this approach to the Toda field theories and the generalized KdV and modified KdV (mKdV) hierarchies. In this paper we apply our approach to the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy [1] and its generalizations. In particular, we show that the free field (Wakimoto) realization of an affine algebra [36, 16] naturally appears in the context of the generalized AKNS hierarchies. This is analogous to the appearance of the free field (quantum Miura) realization of a W–algebra in the context of the generalized KdV equations. As an application, we give here a new proof of the existence of the Wakimoto realization. We also conjecture that all integrals of motion of the generalized AKNS equation can be quantized. In the case of ŝl2 the corresponding quantum integrals of motion can be viewed as integrals of motion of a thermal perturbation of the parafermionic conformal field theory [14]. Thus we expect that this deformation, and analogous deformations for arbitrary affine algebras, are integrable, in the sense of Zamolodchikov [37]. Let us first recall the main steps of our analysis of the Toda field theories from [20, 21].