Algebras of the Virasoro, Neveu-Schwarz, and Ramond Types on Genus g Riemann Surfaces

During the last few years much work has been devoted to trying to reconcile the operator formalism in conformal field theory and string theory, which was originally formulated in the complex plane, with the fact that in Polyakov string theory Riemann surfaces of arbitrary genus must be taken into account. Also, independently of any string interpretation, it is interesting to know the features of a conformal field theory on a Riemann surface of genus g, and therefore it is important to have a manageable operator formalism for any genus. In two recent papers [1, 2] Krichever and Novikov have introduced a new general formalism that may prove very important in this sense. The basic ingredient in their approach is a discrete basis for the algebra of meromorphic vector fields over a Riemann surface that are holomorphic outside two distinguished points. The basis elements form a closed algebra, which is referred to as the Krichever-Novikov (KN) algebra.