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arXiv · 2209.15371

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

Fenglong You
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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.

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Fenglong You. 2022-09-30. The proper Landau--Ginzburg potential, intrinsic mirror symmetry and the relative mirror map. https://doi.org/10.1007/s00220-024-04954-3

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