Conformal and current algebras on a general Riemann surface

Starting from a path integral formulation, Ward identities are derived for conformal and current algebras on a general Riemann surface. An n+1-point amplitude with energy-momentum tensor insertion is related by the Ward identity to an n-point amplitude and its derivative with respect to the modular parameters. Similarly, an n+1-point function with current insertion is related to an n-point function and its derivative with respect to an harmonic background gauge field. We discuss how conformal and current algebras, defined in each coordinate patch, are glued together to form a global algebraic structure on the Riemann surface. (orig.)