Modularity of Landau-Ginzburg models

TL;DR

This paper constructs Landau-Ginzburg models for Fano threefolds that satisfy key mirror symmetry properties, including duality of fibers and Laurent polynomial potentials, through deformation theory and case analysis.

Contribution

It provides a systematic construction of Landau-Ginzburg models for Fano threefolds with detailed verification of mirror symmetry expectations.

Findings

Models are log Calabi-Yau with proper potential maps

Fibers are Dolgachev-Nikulin dual to anticanonical hypersurfaces

Potential functions are Laurent polynomials satisfying Minkowski-type conditions

Abstract

For each Fano threefold, we construct a family of Landau-Ginzburg models which satisfy many expectations coming from different aspects of mirror symmetry; they are log Calabi-Yau varieties with proper potential maps; they admit open algebraic torus charts on which the potential function $w$ restricts to a Laurent polynomial satisfying a deformation of the Minkowski ansatz; the general fibres of $w$ are Dolgachev-Nikulin dual to the anticanonical hypersurfaces in $X$. To do this, we study the deformation theory of Landau-Ginzburg models in arbitrary dimension, following the third-named author, Kontsevich, and Pantev, specializing to the case of Landau-Ginzburg models obtained from Laurent polynomials. Our proof of Dolgachev-Nikulin mirror symmetry is by detailed case-by-case analysis, refining work of Cheltsov and the fifth-named author.