Generalized symmetries andw∞algebras in three-dimensional Toda field theory

Abstract

After establishing a formal theory for getting solutions of one type of high-dimensional partial differential equation, two sets of generalized symmetries of the 3D Toda theory, which arises from a particular reduction of the 4D self-dual gravity equation, are obtained concretely by a simple formula. Each set of symmetries constitutes a generalized ${\mathrm{\omega }}_{\mathrm{\infty }}$ algebra which contains three types of the usual ${\mathrm{\omega }}_{\mathrm{\infty }}$ algebras as special cases. Some open questions are discussed.