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arXiv · 2204.07858

Mirror symmetry for quadric hypersurfaces

Xiaowen Hu
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Abstract

We compute Przyjalkowski-Shramov's resolution of the Calabi-Yau compactification of Givental's mirror Landau-Ginzburg model of the quadric hypersurfaces. We deduce the Picard-Fuchs equation for the narrow periods, which mirror the ambient quantum cohomology of quadric hypersurfaces. Then by an indirect approach using the irreducibility of the narrow Picard-Fuchs operator we deduce the Picard-Fuchs equation of the broad period, which mirrors the quantum cohomology of quadric hypersurfaces involving primitive cohomology classes. The result suggests a natural choice of the opposite space in Barannikov's construction of Frobenius manifolds. Finally, we show an isomorphism between the Frobenius manifolds associated with the quantum cohomology of a quadric hypersurface and its mirror Landau-Ginzburg model.

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Xiaowen Hu. 2022-04-16. Mirror symmetry for quadric hypersurfaces. https://arxiv.org/abs/2204.07858

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