Mirror symmetry for quadric hypersurfaces

TL;DR

This paper computes the mirror Landau-Ginzburg model for quadric hypersurfaces, deriving Picard-Fuchs equations that mirror their quantum cohomology, and establishes an isomorphism between their Frobenius manifolds.

Contribution

It provides explicit computations of mirror symmetry for quadric hypersurfaces, including Picard-Fuchs equations and Frobenius manifold isomorphisms, advancing understanding of mirror symmetry in this context.

Findings

Derived Picard-Fuchs equations for narrow and broad periods.

Established isomorphism between quantum cohomology and mirror Frobenius manifolds.

Suggested a natural choice of opposite space in Frobenius manifold construction.

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.