Finding mirrors for Fano quiver flag zero loci

One interesting feature of the classification of smooth Fano varieties up to dimension three is that they can all be described as certain subvarieties in GIT quotients; in particular, they are all either toric complete intersections (subvarieties of toric varieties) or quiver flag zero loci (subvarieties of quiver flag varieties). Fano varieties are expected to mirror certain Laurent polynomials; given such a Fano toric complete intersection, one can produce a Laurent polynomial via the Landau-Ginzburg model. In this talk, I’ll discuss finding mirrors of four dimensional Fano quiver flag zero loci via finding degenerations of the ambient quiver flag varieties. These degenerations generalise the Gelfand-Cetlin degeneration, which in the Grassmannian case has an important role in the cluster structure of its coordinate ring.