Logarithmic Frobenius manifolds, hypergeometric systems and quantum 𝒟-modules

Reichelt, Thomas; Sevenheck, Christian

doi:10.1090/S1056-3911-2014-00625-1

Logarithmic Frobenius manifolds, hypergeometric systems and quantum $\mathcal D$-modules

Authors: Thomas Reichelt and Christian Sevenheck
Journal: J. Algebraic Geom. 24 (2015), 201-281
DOI: https://doi.org/10.1090/S1056-3911-2014-00625-1
Published electronically: February 12, 2014
MathSciNet review: 3311584
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Abstract | References | Additional Information

Abstract: We describe mirror symmetry for weak Fano toric manifolds as an equivalence of filtered $\mathcal {D}$-modules. 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 $\mathcal {D}$-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 Kähler moduli space of the variety.

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Additional Information

Thomas Reichelt
Affiliation: Lehrstuhl für Mathematik VI, Institut für Mathematik, Universität Mannheim, A 5, 6, 68131 Mannheim, Germany
Email: Thomas.Reichelt@math.uni-mannheim.de

Christian Sevenheck
Affiliation: Lehrstuhl für Mathematik VI, Institut für Mathematik, Universität Mannheim, A 5, 6, 68131 Mannheim, Germany
Email: Christian.Sevenheck@math.uni-mannheim.de

Received by editor(s): August 12, 2011
Received by editor(s) in revised form: September 20, 2012
Published electronically: February 12, 2014
Additional Notes: The first author was supported by the DFG grant He 2287/2-2.
The second author was supported by a DFG Heisenberg fellowship (Se 1114/2-1).
Article copyright: © Copyright 2014 University Press, Inc.
The copyright for this article reverts to public domain 28 years after publication.