Open FJRW Theory and Mirror Symmetry

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.

143 pages, minor revisions, to appear in Geometry and Topology