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Generalized symmetries and \(w_\infty\) algebras in three-dimensional Toda field theory. (English) Zbl 0972.81559

Summary: 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 \(w_\infty\) algebra which contains three types of the usual \(w_\infty\) algebras as special cases. Some open questions are discussed.

MSC:

81R20 Covariant wave equations in quantum theory, relativistic quantum mechanics
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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