Mirror Symmetry and Supermanifolds

We develop techniques for obtaining the mirror of Calabi-Yau supermanifolds as super Landau-Ginzburg theories. In some cases the dual can be equivalent to a geometry. We apply this to some examples. In particular we show that the mirror of the twistorial Calabi-Yau CP becomes equivalent to a quadric in CP×CP as had been recently conjectured (in the limit where the Kähler parameter of CP, t → ±∞). Moreover we show using these techniques that there is a non-trivial Z2 symmetry for the Kähler parameter, t → −t, which exchanges the opposite helicity states. As another class of examples, we show that the mirror of certain weighted projective (n+ 1|1) superspaces is equivalent to compact Calabi-Yau hypersurfaces in weighted projective n space.