Logarithmic Frobenius manifolds, hypergeometric systems and quantum D-modules

By Thomas Reichelt, Christian Sevenheck. 2010

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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 Gaus-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

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9747a6b0ab7b7551d45ac29227bb3f1c2b159395

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Citations in the corpus

The Crepant Transformation Conjecture for Toric Complete Intersections (2014, background): “In the papers [51,71], the toric quantum connection is described in terms of GKZ-type differential equations through mirror symmetry.”