Mirror Symmetry and Supermanifolds

TL;DR

This paper develops methods to find mirror theories of Calabi-Yau supermanifolds, revealing geometric duals and symmetries, with applications to twistorial spaces and weighted projective superspaces.

Contribution

It introduces techniques for deriving mirror super-Landau-Ginzburg models, demonstrating geometric equivalences and symmetries in Calabi-Yau supermanifolds.

Findings

Mirror of CP^{3|4} is equivalent to a quadric in CP^{3|3} x CP^{3|3}.

Identifies a Z_2 symmetry exchanging helicity states via t -> -t.

Mirror of weighted projective superspaces corresponds to Calabi-Yau hypersurfaces.

Abstract

We develop techniques for obtaining the mirror of Calabi-Yau supermanifolds as super-Landau-Ginzburg theories. In some cases the dual can be equivalent to a geometry. We apply this to some examples. In particular we show that the mirror of the twistorial Calabi-Yau CP^{3|4} becomes equivalent to a quadric in CP^{3|3} x CP^{3|3} as had been recently conjectured (in the limit where the K\"ahler parameter of CP^{3|4} t -> \pm \infty). Moreover, we show using these techniques that there is a non-trivial Z_2 symmetry for the K\"ahler parameter, t -> -t, which exchanges the opposite helicity states. As another class of examples, we show that the mirror of certain weighted projective (n+1|1) superspaces is equivalent to compact Calabi-Yau hypersurfaces in weighted projective n-space.