Open FJRW Theory and Mirror Symmetry

TL;DR

This paper develops an open enumerative theory for a specific Landau-Ginzburg model, establishing mirror symmetry results, wall-crossing phenomena, and a recursion formula in dimension two.

Contribution

It introduces a new open invariants framework for LG models, proves an open mirror symmetry theorem, and describes wall-crossing transformations.

Findings

Constructed open invariants for LG models.

Proved open LG/LG mirror symmetry in dimension two.

Described wall-crossing group for invariants.

Abstract

We construct an open enumerative theory for the Landau-Ginzburg (LG) model $(\mathbb{C}^2, \mu_r\times \mu_s, x^r+y^s)$. The invariants are defined as integrals of multisections of a Witten bundle with descendents over a moduli space that is a real orbifold with corners. In turn, a generating function for these open invariants yields the mirror LG model and a versal deformation of it with flat coordinates. After establishing an open topological recursion result, we prove an LG/LG open mirror symmetry theorem in dimension two with all descendents. The open invariants we define are not unique but depend on boundary conditions that, when altered, exhibit wall-crossing phenomena for the invariants. We describe an LG wall-crossing group classifying the wall-crossing transformations that can occur.