Logarithmic Frobenius manifolds, hypergeometric systems and quantum D-modules
Logarithmic Frobenius manifolds, hypergeometric systems and quantum D-modules
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DOI：
10.1090/s1056-3911-2014-00625-1
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发表时间：
2010-10
影响因子：
1.8
通讯作者：
Thomas Reichelt;Christian Sevenheck
中科院分区：
文献类型：
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作者：
Thomas Reichelt;Christian Sevenheck
我们将弱环形范诺流形的镜像对称性描述为具有特定滤光的d模的等价。我们特别讨论了大半径极限点处的对数退化行为，并将镜像对应表示为具有对数极点的Frobenius流形的同构。本文的主要工具是用超几何d模对镜面Landau-Ginzburg模型的高斯-曼系统进行识别，并详细研究了在该微分系统上定义的自然滤波。我们得到了由这种过滤定义的格子的Birkhoff问题的一个解，并证明了一个原始形式的存在，它产生了与镜像Laurent多项式相关的对数极点的Frobenius结构的构造。作为最后一个应用，我们证明了a上存在一个纯极化非交换Hodge结构
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