Complex structures, duality and WZW-models in extended superspace

We find the complex structure on the dual of a complex target space. For N = (2,2) systems, we prove that the space orthogonal to the kernel of the commutator of the left and right complex structures is always integrable, and hence the kernel is parametrized by chiral and twisted chiral superfield coordinates. We then analyze the particular case of SU(2) × SU(2), and are led to a new N = 2 superspace formulation of the SU(2) × U(1) WZW-model.