Worldsheets with extended supersymmetry

We determine the equations which govern the gauge symmetries of worldsheets with local supersymmetry of arbitrary rank (N, N′), and their possible anomalies. Both classical and ghost conformally invariant multiplets of the left or right sector are assembled into the components of a single O(N)-superfield. The component with ghost number zero of this superfield is the N-supersymmetric generalization of the Beltrami differential. In a Lagrangian approach, and after gauge-fixing, it becomes the super-moduli of Riemann surfaces coupled to local supersymmetry of rank N. It is also the source of all linear superconformal currents derived from ordinary operator product techniques. The interconnection between BRST invariant actions with different values of N ⩾ 3, and their possible link to topological 2D-gravity coupled to topological sigma models, are shown by straightforward algebraic considerations.