The B-model connection and mirror symmetry for Grassmannians
We consider the Grassmannian X=Grn−k(Cn) and describe a ‘mirror dual’ Landau-Ginzburg model (Xˇ∘,Wq:Xˇ∘→C), where Xˇ∘ is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian Xˇ, and we express W succinctly in terms of Plücker coordinates. First of all, we show this...
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Published in | Advances in mathematics (New York. 1965) Vol. 366; p. 107027 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Elsevier Inc
03.06.2020
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Subjects | |
Online Access | Get full text |
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Summary: | We consider the Grassmannian X=Grn−k(Cn) and describe a ‘mirror dual’ Landau-Ginzburg model (Xˇ∘,Wq:Xˇ∘→C), where Xˇ∘ is the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian Xˇ, and we express W succinctly in terms of Plücker coordinates. First of all, we show this Landau-Ginzburg model to be isomorphic to one proposed for homogeneous spaces in a previous work by the second author. Secondly we show it to be a partial compactification of the Landau-Ginzburg model defined in the 1990's by Eguchi, Hori, and Xiong. Finally we construct inside the Gauss-Manin system associated to Wq a free submodule which recovers the trivial vector bundle with small Dubrovin connection defined out of Gromov-Witten invariants of X. We also prove a T-equivariant version of this isomorphism of connections. Our results imply in the case of Grassmannians an integral formula for a solution to the quantum cohomology D-module of a homogeneous space, which was conjectured by the second author. They also imply a series expansion of the top term in Givental's J-function, which was conjectured in a 1998 paper by Batyrev, Ciocan-Fontanine, Kim and van Straten. |
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ISSN: | 0001-8708 1090-2082 |
DOI: | 10.1016/j.aim.2020.107027 |