Supersymmetric chiral boson with linearly realized self-duality condition

Two supersymmetric versions for the theory of a chiral boson with linearly realized self-duality condition are proposed. In both cases the sector of physical states after canonical quantization contains only modes propagating to the left. The first version describes a chiral boson and its superpartner; the second does not contain nontrivial Fermi degrees of freedom. Both actions are rewritten in a locally supersymmetric form by introducing an interaction with $(1,0)$ supergravity in the superfield and component forms.