The B-model connection and mirror symmetry for Grassmannians

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 Pl\"ucker 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.