One-Dimensional Super Calabi-Yau Manifolds and their Mirrors

TL;DR

This paper explores one-dimensional super Calabi-Yau manifolds, computes their cohomology, automorphism groups, and deformations, and investigates their mirror symmetry properties, revealing unique features of specific supermanifolds.

Contribution

It introduces a detailed analysis of super Calabi-Yau varieties of complex dimension one, including cohomology, automorphisms, deformations, and mirror symmetry, with explicit examples and new findings.

Findings

Identified two super Calabi-Yau manifolds with reduced space $P^1$.

Computed sheaf cohomology showing infinite-dimensionality for certain forms.

Established mirror symmetry properties, including self-mirror and zero-dimensional 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 $\mathbb{WP}^{1|1}_{(2)}$. 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, 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 $\mathbb{WP} ^{1|1}_{(2)}$ 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.