Toda Theories [formula omitted]-algebras, and minimal models

Abstract

We discuss the classical W -algebra symmetries of Toda field theories in terms of the pseudo-differential Lax operator associated with the Toda Lax pair. We then show how the W -algebra transformations can be understood as the non-abelian gauge transformations which preserve the form of the Lax pair. This provides a new understanding of the W -algebras, and we discuss their closure and co-cycle structure using this approach. The quantum Lax operator is investigated, and we show that this operator, which generates the quantum W -algebra currents, is concerned in the conformally extended Toda theories. The W -algebra minimal model primary fields are shown to arise naturally in these theories, leading to the conjecture that the conformally extended Toda theories provide a lagrangian formulation of the W -algebra minimal models.