One dimensional super Calabi Yau manifolds and their mirrors
A bstractWe 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 P1$ \mathrm{\mathbb{P}^1 namely the projective super space P12$ \mathrm{\mathbb{P}\left.1\right|2} and the weighted projective super space WP211$ \mathbb{W}\mathrm{\mathbb{P}}_{(2)\left.1\right|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 Pnm$ \mathrm{\mathbb{P}\left.n\right|m} We also determine the automorphism groups: these always match the dimension of the projective super group with the only exception of P12$ \mathrm{\mathbb{P}\left.1\right|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 P1m$ \mathrm{\mathbb{P}\left.1\right|m} discovering that the presence of a fermionic structure allows for deformations even if the reduced manifold is rigid. Finally, we show that P12$ \mathrm{\mathbb{P}\left.1\right|2} is self mirror, whereas WP211$ \mathbb{W}\mathrm{\mathbb{P}}_{(2)\left.1\right|1} has a zero dimensional mirror. Also, the mirror map for P12$ \mathrm{\mathbb{P}\left.1\right|2} naturally endows it with a structure of N 2 super Riemann surface.