New Linear Systems for 2D Poincare Supergravities

A new linear system is constructed for Poincar\'e supergravities in two dimensions. In contrast to previous results, which were based on the conformal gauge, this linear system involves the topological world sheet degrees of freedom (the Beltrami and super-Beltrami differentials). The associated spectral parameter likewise depends on these and is itself subject to a pair of differential equations, whose integrability condition yields one of the equations of motion. These results suggest the existence of an extension of the Geroch group mixing propagating and topological degrees of freedom on the world sheet. We also develop a chiral tensor formalism for arbitrary Beltrami differentials, in which the factorization of $2d$ diffeomorphisms is always manifest.