Mirror symmetry for honeycombs

Abstract

We prove a homological mirror symmetry equivalence between the A-brane category of the pair of pants, computed as a wrapped microlocal sheaf category, and the B-brane category of its mirror LG model, understood as a category of matrix factorizations. The equivalence improves upon prior results in two ways: it intertwines evident affine Weyl group symmetries on both sides, and it exhibits the relation of wrapped microlocal sheaves along different types of Lagrangian skeleta for the same hypersurface. The equivalence proceeds through the construction of a combinatorial realization of the A-model via arboreal singularities. The constructions here represent the start of a program to generalize to higher dimensions many of the structures which have appeared in topological approaches to Fukaya categories of surfaces.

Tailoring a pair of pants

“…On the symplectic side Sheridan [Sh11] first made use of the zonotope, the complement of the coamoeba, by viewing its boundary as an immersed Lagrangian sphere in the pair-of-pants. More recently, Nadler-Gammage [GN20] were able to view the skeleton of the coamoeba as a Lagrangian. A different Lagrangian skeleton had previously been given by [RSTZ,Zh18].…”