Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1994-03-07
J.Geom.Phys. 16 (1995) 237-274
Physics
High Energy Physics
High Energy Physics - Theory
38 pages, one relevant reference has been added
Scientific paper
10.1016/0393-0440(94)00028-3
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.
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