Abstract
A general class of conformal Toda theories associated with integral gradings of the simple Lie algebras is investigated. These generalized Toda theories are obtained by reducing the Wess-Zumino-Novikov-Witten (WZNW) theory by first class constraints, and thus they inherite extended conformal symmetry algebras, generalized W -algebras, and current dependent Kac-Moody (KM) symmetries from the WZNW theory, which are analysed in detail in a non-degenerate case. We uncover an sl(2) structure underlying the generalized W -algebras, which allows for identifying the primary fields, and give a simple algorithm for implementing the W -symmetries by current dependent KM transformations, which can be used to compute the action of the W -algebra on any quantity. We establish how the Lax pair of Toda theory arises in the WZNW framework and show that a recent result of Mansfield and Spence, which interprets the W -symmetry of the Toda theory by means of non-Abelian form preserving gauge transformations of the Lax pair, arises immediately as a consequence of the KM interpretation.
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