Geometric Interpretation of the Poisson Structure in Affine Toda Field Theories Benjamin Enriquez and Edward Frenkel

We express the Poisson brackets of local fields of the affine Toda field theories in terms of the Drinfeld-Sokolov dressing operator. For this, we introduce a larger space of fields, containing “half screening charges” and “half integrals of motions”. In addition to local terms, the Poisson brackets contain nonlocal terms related to trigonometric r-matrices. Introduction Since the work of Zakharov and Shabat [23], the dressing techniques have played an important role in the theory of classical integrable systems. These techniques have been developed by Drinfeld and Sokolov in [7] in the framework of affine Toda field theories. Later, Feigin and one of us proposed ([13], [14]) another approach to these theories; this approach was shown ([8], [10]) to be equivalent to that of [7]. In those works, the space of local fields of the Toda theory (equivalently, the mKdV hierarchy) associated to an affine Lie algebra g is described as the ring of functions on the coset space N+/A+ of a unipotent subgroup of the Kac-Moody group G corresponding to g. The mKdV flows are then identified with the right action of the principal commutative Lie algebra a normalizing A+, N+ being viewed as an open subset of the flag manifold of G. This leads to a system of variables, in which the flows become linear and hence can be integrated. In the works on quantization of the Toda theories, an important role is played by the vertex operator algebra structure on the space of local fields. At the classical level, this gives rise to what we call here a vertex Poisson algebra (VPA) structure on the space of local fields of a Toda theory. The notion of the VPA structure coincides with the notion of “coisson algebra” (on the disc) introduced by Beilinson and Drinfeld in [5]. The goal of this work is to define this and related structures on the space of fields of a Toda theory in the Lie group terms using the identification described above. The research of the second author was partially supported by grants from the Packard Foundation, NSF and the Sloan Foundation