The Drinfeld-Sokolov holomorphic bundle and classical W algebras on Riemann surfaces

Developing upon the ideas of an earlier publication it is shown how the theory of classical W algebras can be formulated on a higher genus Riemann surface in the spirit of Krichever and Novikov. The basic geometric object is the Drinfeld-Sokolov principal bundle L associated to a simple complex Lie group G equipped with an SL (2, C) subgroup S, whose properties are studied in detail. On a multipunctured Riemann surface, the Drinfeld-Sokolov-Krichever-Novikov spaces are defined as a generalization of the customary Krichever-Novikov spaces, their properties are analyzed and standard bases are written down. Finally, a WZWN chiral phase space based on the principal bundle L with a KM type Poisson structure is introduced and, by the usual procedure of imposing first class constraints and gauge fixing, a classical W algebra is produced. The compatibility of the construction with the global geometric data is highlighted. © 1995.