The B-model connection and mirror symmetry for Grassmannians

Robert Marsh, Konstanze Rietsch

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We consider the Grassmannian X=Gr n−k(C n) and describe a ‘mirror dual’ Landau-Ginzburg model (Xˇ ,W q: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 W q 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.

Original languageEnglish
Article number107027
Pages (from-to)1-131
Early online date5 Mar 2020
Publication statusPublished - 3 Jun 2020


  • Cluster algebras
  • Gauss-Manin system
  • Grassmannian quantum cohomology
  • Gromov-Witten theory
  • Landau-Ginzburg model
  • Mirror symmetry


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