Open FJRW Theory and Mirror Symmetry

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.