CONFORMALLY COVARIANT OPERATORS ON RIEMANN SURFACES (WITH APPLICATIONS TO CONFORMAL AND INTEGRABLE MODELS)

Following the standard procedure for gauging in Yang-Mills and gravitational theories, we introduce projective connections to covariantize differential operators on Riemann surfaces. We present applications in integrable models (Lax pairs, Poisson operators) and conformal models (conformal Ward identity, diffeomorphism anomaly, Krichever-Novikov algebra, Virasoro algebra and its representations, Kac-Moody algebras, W-algebras, WZW model, twistor theory). The generalization to higher dimensions is indicated and the whole discussion is generalized to the supersymmetric case (both in superspace and in component field formalism).