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Mirror symmetry and supermanifolds. (English) Zbl 1094.32006

Summary: We develop techniques for obtaining the mirror of Calabi-Yau super-manifolds 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-Yan \(\mathbb{C}\mathbb{P}^{3|4}\) becomes equivalent to a quadric in \(\mathbb{C}\mathbb{P}^{3|3}\times\mathbb{C}\mathbb{P}^{3|3}\) as had been recently conjectured (in the limit where the Kähler parameter of \(\mathbb{C}\mathbb{P}^{3|4},t\to\pm \infty)\). Moreover we show using these techniques that there is a non-trivial \(\mathbb{Z}_2\) symmetry for the Kähler parameter, \(t\to-t\), 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.

MSC:

32L25 Twistor theory, double fibrations (complex-analytic aspects)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
32Q25 Calabi-Yau theory (complex-analytic aspects)
32C11 Complex supergeometry
81T60 Supersymmetric field theories in quantum mechanics
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