The proper Landau--Ginzburg potential, intrinsic mirror symmetry and the relative mirror map

Abstract

Given a smooth log Calabi--Yau pair $(X,D)$, we use the intrinsic mirror symmetry construction to define the mirror proper Landau--Ginzburg potential and show that it is a generating function of two-point relative Gromov--Witten invariants of $(X,D)$. We compute certain relative invariants with several negative contact orders, and then apply the relative mirror theorem of \cite{FTY} to compute two-point relative invariants. When $D$ is nef, we compute the proper Landau--Ginzburg potential and show that it is the inverse of the relative mirror map. Specializing to the case of a toric variety $X$, this implies the conjecture of \cite{GRZ} that the proper Landau--Ginzburg potential is the open mirror map. When $X$ is a Fano variety, the proper potential is related to the anti-derivative of the regularized quantum period.