We consider the Grassmannian X of (n-k)-dimensional subspaces of an n-dimensional complex vector space. We describe a `mirror dual' Landau-Ginzburg model for X consisting of the complement of a particular anti-canonical divisor in a Langlands dual Grassmannian together with a superpotential expressed succinctly in terms of Plucker coordinates. First of all, we show this Landau-Ginzburg model to be isomorphic to the one proposed by the second author. Secondly we show it to be a partial compactification of the Landau-Ginzburg model defined in the 1990s by Eguchi, Hori, and Xiong. Finally we construct inside the Gauss-Manin system associated to the superpotential 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-Fontaine, Kim and van Straten.

中文翻译：

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Grassmannians 的 B 模型连接和镜像对称

我们考虑一个 n 维复向量空间的 (nk) 维子空间的 Grassmannian X。我们描述了 X 的“镜像对偶”Landau-Ginzburg 模型，该模型由朗兰兹对偶 Grassmannian 中的特定反规范除数的补集以及用 Plucker 坐标简明表达的超势组成。首先，我们展示了这个 Landau-Ginzburg 模型与第二作者提出的模型同构。其次，我们证明它是 Eguchi、Hori 和 Xiong 在 1990 年代定义的 Landau-Ginzburg 模型的部分紧缩。最后，我们在与超势相关的 Gauss-Manin 系统内部构造了一个自由子模块，该子模块恢复具有由 X 的 Gromov-Witten 不变量定义的小 Dubrovin 连接的平凡向量丛。我们还证明了这种连接同构的 T 等变版本。我们的结果在 Grassmannians 的情况下暗示了解决齐次空间的量子上同调 D 模的积分公式，这是由第二作者推测的。它们还暗示了 Givental 的 J 函数中顶项的一系列展开，这是由 Batyrev、Ciocan-Fontaine、Kim 和 van Straten 在 1998 年的一篇论文中推测的。