# One-dimensional super Calabi-Yau manifolds and their mirrors

Journal of High Energy Physics, Apr 2017

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

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S. Noja, S. L. Cacciatori, F. Dalla Piazza, A. Marrani, R. Re. One-dimensional super Calabi-Yau manifolds and their mirrors, Journal of High Energy Physics, 2017, 94, DOI: 10.1007/JHEP04(2017)094