One-dimensional super Calabi-Yau manifolds and their mirrors

We apply a definition of generalised super Calabi-Yau variety (SCY) to supermanifolds of complex dimension one. One of our results is that there are two SCY’s having reduced manifold equal to ℙ 1, namely the projective super space ℙ 1 | 2 and the weighted projective super space Wℙ(2)1|1. Then we compute the corresponding sheaf cohomology of superforms, showing that the cohomology with picture number one is infinite dimensional, while the de Rham cohomology, which is what matters from a physical point of view, remains finite dimensional. Moreover., we provide the complete real and holomorphic de Rham cohomology for generic projective super spaces ℙ n|m. We also determine the automorphism groups: these always match the dimension of the projective super group with the only exception of ℙ 1 | 2, whose automorphism group turns out to be larger than the projective super group. By considering the cohomology of the super tangent sheaf, we compute the deformations of ℙ 1|m, discovering that the presence of a fermionic structure allows for deformations even if the reduced manifold is rigid. Finally., we show that ℙ 1 | 2 is self-mirror, whereas Wℙ(2)1|1 has a zero dimensional mirror. Also., the mirror map for ℙ 1 | 2 naturally endows it with a structure of N = 2 super Riemann surface.