# Mirror symmetry for honeycombs

## TL;DR

This paper establishes a homological mirror symmetry equivalence between an A-brane category for the pair of pants and a B-brane category for a mirror Landau-Ginzburg model, highlighting symmetries and Lagrangian skeleta.

## Contribution

It advances mirror symmetry by intertwining affine Weyl group symmetries and relating microlocal sheaves along different Lagrangian skeleta, using combinatorial arboreal singularities.

## Key findings

- Proves a mirror symmetry equivalence for the pair of pants.
- Intertwines affine Weyl group symmetries on both sides.
- Relates microlocal sheaves along different Lagrangian skeleta.

## Abstract

We prove a homological mirror symmetry equivalence between an $A$-brane category for the pair of pants, computed as a wrapped microlocal sheaf category, and a $B$-brane category for a 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.