Wall-Crossing in Genus Zero Landau-Ginzburg Theory

Abstract

We study genus zero wall-crossing for a family of moduli spaces introduced recently by Fan-Farvis-Ruan. The family has a wall and chamber structure relative to a positive rational parameter. For a Fermat quasi-homogeneous polynomial W (not necessarily Calabi-Yau type), we study natural generating functions of invariants associated to these moduli spaces. Our wall-crossing formula relates the generating functions by showing that they all lie on the same Lagrangian cone associated to the Fan-Jarvis-Ruan-Witten theory of W. For arbitrarily small parameter, a specialization of our generating function is a hypergeometric series called the big I-function which determines the entire Lagrangian cone. As a special case of our wall-crossing, we obtain a new geometric interpretation of the Landau-Ginzburg mirror theorem.