Categorification and the quantum Grassmannian

In an earlier work, we gave an (additive) categorification of Grassmannian cluster algebras using the category CM(A) of Cohen-Macaulay modules for a certain Gorenstein order A. In this paper, for each cluster tilting object in CM(A), we construct a compatible pair (B,L) in a way that is consistent with mutation. This then determines a quantum cluster algebra and we show that, when (B,L) comes from a cluster tilting object with rank one summands, this quantum cluster algebra is (generically) isomorphic to the corresponding quantum Grassmannian. The key ingredient in the construction is a new invariant κ(M,N) of modules M,N in CM(A), which also has an intriguing link with mirror symmetry.