WORLDSHEETS WITH EXTENDED SUPERSYMMETRY

We determine the equations which govern the gauge symmetries of worlds heets with local supersymmetry of arbitrary rank (N, N'), and their po ssible anomalies. Both classical and ghost conformally invariant multi plets of the left or right sector are assembled into the components of a single O(N)-superfield. The component with ghost number zero of thi s superfield is the N-supersymmetric generalization of the Beltrami di fferential. In a Lagrangian approach, and after gauge-fixing, it becom es the super-moduli of Riemann surfaces coupled to local supersymmetry of rank N. It is also the source of all linear superconformal current s derived from ordinary operator product techniques, The interconnecti on between BRST invariant actions with different values of N greater t han or equal to 3, and their possible link to topological 2D-gravity c oupled to topological sigma models, are shown by straightforward algeb raic considerations.