Exotic Lagrangian tori in Grassmannians

Abstract

For each plabic graph of type (k,n) in the sense of Postnikov satisfying a smallness condition, we construct a nondisplaceable monotone Lagrangian torus in the complex Grassmannian Gr(k,n). Among these we find examples that bound the same number of families of Maslov 2 pseudoholomorphic disks, whose Hamiltonian isotopy classes are distinguished by the number of critical points in different algebraic torus charts of a mirror Landau-Ginzburg model proposed by Marsh-Rietsch. The tori are fibers of local regular Lagrangian fibrations over Okounkov bodies for the frozen anticanonical divisor, which is singled out by the cluster structure of the Grassmannian and has been studied by Rietsch-Williams. Lagrangian tori of plabic graphs related by a combinatorial square move have disk potentials connected by a 3-term Plucker relation, while their Newton polytopes undergo width 2 mutation in the sense of Akhtar-Coates-Galkin-Kasprzyk.