One-dimensional super Calabi-Yau manifolds and their mirrors

Abstract

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

Keywords

• Differential and Algebraic Geometry
• Superspaces
• String Duality
• Supersymmetry and Duality