One-Dimensional Super Calabi-Yau Manifolds and their Mirrors

Abstract

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 ${\mathbb{P}}^1$, namely the projective super space ${\mathbb{P}}^{1|2}$ and the weighted projective super space ${\mathbb{WP}}_{(2)}^{1|1}$. 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 ${\mathbb{P}}^{n|m}$. We also determine the automorphism groups: these always match the dimension of the projective super group with the only exception of ${\mathbb{P}}^{1|2}$, whose automorphism group turns out to be larger than the projective general linear supergroup. By considering the cohomology of the super tangent sheaf, we compute the deformations of ${\mathbb{P}}^{1|m}$, discovering that the presence of a fermionic structure allows for deformations even if the reduced manifold is rigid. Finally, we show that ${\mathbb{P}}^{1|2}$ is self-mirror, whereas ${\mathbb{WP}}_{(2)}^{1|1}$ has a zero dimensional mirror. Also, the mirror map for ${\mathbb{P}}^{1|2}$ naturally endows it with a structure of $N=2$ super Riemann surface.

arXiv comment50 pages. Accepted for publication in JHEP

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• $A_{\infty }$-Algebra from Supermanifolds
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• Superstring Field Theory, Superforms and Supergeometry
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• Non-Projected Supermanifolds and Embeddings in Super Grassmannians