Mirror symmetry for Grassmannians

We will write down a Landau-Ginzburg model for an arbitrary Grassmannian X in terms of Pluecker coordinates on a dual Grassmannian. This LG-model gives rise to a vector bundle with Gauss-Manin connection, and we relate this vector bundle explicitly with a bundle on the A-model side of X: the trivial bundle over C with fibre H*(X) and its small Dubrovin-Givental connection. To show that our map between the vector bundles takes one flat connection to the other we make heavy use of the cluster structure of the homogeneous coordinate ring of the dual Grassmannian.