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

TL;DR

This paper constructs the mirror Landau--Ginzburg potential for log Calabi--Yau pairs using intrinsic mirror symmetry, relates it to relative Gromov--Witten invariants, and confirms conjectures connecting it to mirror maps, especially in toric and Fano cases.

Contribution

It defines the proper Landau--Ginzburg potential via intrinsic mirror symmetry and proves it equals the inverse of the relative mirror map for nef divisors, confirming conjectures in toric and Fano cases.

Findings

Proper Landau--Ginzburg potential is a generating function of two-point relative Gromov--Witten invariants.

When D is nef, the potential equals the inverse of the relative mirror map.

In toric and Fano cases, the potential matches the open mirror map and relates to the quantum period.

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.