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We explicitly construct bases for meromorphic λ-differentials over genus g Riemann surfaces. With the help of these bases we introduce a new operator formalism over Riemann surfaces which closely resembles the operator formalism on the sphere. As an application we calculate the propagators for b-c systems with arbitrary integer or half-integer λ (in the Ramond and Neveu-Schwarz sectors). We also give explicit expressions for the zero modes and for the Teichmüller deformations for a generic Riemann surface

A global operator formalism on higher genus Riemann surfaces:b?c systems