Logarithmic Frobenius manifolds, hypergeometric systems and quantum D-modules

Abstract

We describe mirror symmetry for weak toric Fano manifolds as an equivalence of D-modules equipped with certain filtrations. We discuss in particular the logarithmic degeneration behavior at the large radius limit point, and express the mirror correspondence as an isomorphism of Frobenius manifolds with logarithmic poles. The main tool is an identification of the Gauss-Manin system of the mirror Landau-Ginzburg model with a hypergeometric D-module, and a detailed study of a natural filtration defined on this differential system. We obtain a solution of the Birkhoff problem for lattices defined by this filtration and show the existence of a primitive form, which yields the construction of Frobenius structures with logarithmic poles associated to the mirror Laurent polynomial. As a final application, we show the existence of a pure polarized non-commutative Hodge structure on a Zariski open subset of the complexified Kaehler moduli space of the variety.

References in corpus (8)

• An integral structure in quantum cohomology and mirror symmetry for toric orbifolds
• Hypergeometric D-modules and twisted Gauss-Manin systems
• Asymptotic behaviour of variation of pure polarized TERP structures
• A construction of Frobenius manifolds with logarithmic poles and applications
• tt*-geometry in quantum cohomology
• Non-commutative Hodge structures
• The small quantum cohomology of a weighted projective space, a mirror D-module and their classical limits
• $A$-hypergeometric systems that come from geometry

Cited by in corpus (4)

• The Crepant Transformation Conjecture for Toric Complete Intersections
• Hodge-Tate conditions for Landau-Ginzburg models
• Quantum differential systems and some applications to mirror symmetry
• The small quantum cohomology of a weighted projective space, a mirror D-module and their classical limits