## Abstract

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 language | English |
---|---|

Article number | 107027 |

Pages (from-to) | 1-131 |

Journal | ADVANCES IN MATHEMATICS |

Volume | 366 |

Early online date | 5 Mar 2020 |

DOIs | |

Publication status | Published - 3 Jun 2020 |

## Keywords

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