The small quantum cohomology ring of a Grassmannian X encodes enumerative information about holomorphic maps from CP^1 to X, and has a well known presentation and associated combinatorics involving Young diagrams (quantum Schubert calculus). Considering the quantum cup product with the hyperplane class gives rise to a D-module via the small Dubrovin connection. The goal of this talk is to write down a regular function defined on a ‘mirror dual’ Grassmannian with an anti-canonical divisor removed, which we can show encodes the same D-module in the form of a Gauss-Manin system. This regular function also restricts, in suitable coordinates, to the Laurent polynomial superpotential first proposed by Eguchi Hori and Xiong, and our mirror theorem implies a formula for the constant term of the J-function conjectured by Batyrev, Ciocan-Fontanine, Kim and van Straten in 1998. A central role in our proofs (joint work with Robert Marsh) is played by the cluster algebra structure of the homogeneous coordinate ring of the mirror Grassmannian and works of Postnikov, Scott and Marsh-Scott.
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