  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.
