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. We get that the class of all SCY's of bosonic dimension one and reduced manifold equal to ℙ1 is given by ℙ1|2 and the weighted projective super space ℙ1|1(2). 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 remains finite dimensional. Moreover, we provide the complete real and holomorphic de Rham cohomology for generic projective super spaces ℙn|m. We also determine the automorphism groups, which for ℙ1|2 results to be larger than the projective supergroup. Finally, we show that ℙ1|2 is self mirror, whereas ℙ1|1(2) has a zero dimensional mirror. The mirror map for ℙ1|2 endows it with a structure of N=2 super Riemann surface.