The Drinfeld--Sokolov Holomorphic Bundle and Classical $W$ Algebras on Riemann Surfaces

TL;DR

This paper extends the theory of classical W algebras to higher genus Riemann surfaces using Drinfeld--Sokolov bundles, defining new spaces and deriving classical W algebras with geometric consistency.

Contribution

It formulates classical W algebras on higher genus Riemann surfaces via Drinfeld--Sokolov bundles and introduces generalized Krichever--Novikov spaces with explicit bases.

Findings

Defined Drinfeld--Sokolov--Krichever--Novikov spaces

Constructed a WZWN chiral phase space with Poisson structure

Derived classical W algebras compatible with geometric data

Abstract

Developing upon the ideas of ref. \ref{6}, 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 $SL(2,\Bbb C)$ 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.