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In [Rie08], the second author defined a Landau-Ginzburg model for homogeneousspaces G/P, as a regular function on an affine subvariety of the Langlands dualgroup. In this paper, we reformulate this LG-model (X^,W_t) in the case of theLagrangian Grassmannian LG(m) as a rational function on a Langlands dualorthogonal Grassmannian, in the spirit of work by R. Marsh and the secondauthor [MR12] for type A Grassmannians. This LG model has some very interestingfeatures, which are not visible in the type A case, to do with thenon-triviality of Langlands duality.  We also formulate a conjecture relating our superpotential with the quantumdifferential equations of LG(m). Finally, our expression for W_t also leads usto conjecture new formulas in the quantum Schubert calculus of LG(m).

A Landau-Ginzburg model for Lagrangian Grassmannians, Langlands duality and relations in quantum cohomology