Duality Invariant Superstring Actions
  Worldsheet supersymmetric string action is written in duality invariant form for flat as well as curved backgrounds. First the action in flat backgrounds is written by introducing auxiliary fields. We also give the superfield form of this action and obtain the offshell supersymmetry algebra. The action has a modified Lorentz invariance and supersymmetry and reduces to the usual form when the auxiliary fields are eliminated using their equations of motion. Supersymmetric nonlinear sigma model in curved backgrounds is also written in manifestly duality invariant form when the background metric and tortion fields are independent of some of the coordinates. 
Abbas Ali