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 vertical bar 2 and the weighted projective super space WP(2)1 vertical bar 1. Then we compute the corresponding sheaf cohomology of superforms, showing that the cohomology with picture number one is in finite 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 vertical bar m. We also determine the automorphism groups: these always match the dimension of the projective super group with the only exception of P-1 vertical bar 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 vertical bar 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 vertical bar 2 is self-mirror, whereas WP(2)1 vertical bar 1 has a zero dimensional mirror. Also, the mirror map for P-1 vertical bar 2 naturally endows it with a structure of N = 2 super Riemann surface.