Wall-crossing in genus zero Landau–Ginzburg theory

Abstract

Abstract We study a family of moduli spaces and corresponding quantum invariants introduced recently by Fan–Jarvis–Ruan. The family has a wall-and-chamber structure relative to a positive rational parameter ϵ. For a Fermat quasi-homogeneous polynomial W (not necessarily of Calabi–Yau type), we study natural generating functions packaging the invariants. Our wall-crossing formula relates the generating functions by showing that they all lie on the Lagrangian cone associated to the Fan–Jarvis–Ruan–Witten theory of W. For arbitrarily small parameter ϵ, 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.