A Landau--Ginzburg mirror theorem without concavity

TL;DR

This paper establishes a mirror symmetry theorem for FJRW potentials in non-convex cases, providing explicit formulas and compatibility results that extend mirror symmetry beyond traditional convexity assumptions.

Contribution

It introduces a new mirror symmetry theorem applicable to non-convex FJRW potentials, with explicit formulas and compatibility results, expanding the scope of mirror symmetry in quantum singularity theory.

Findings

Explicit formula for genus zero virtual cycle

Compatibility theorem with FJRW virtual cycle

Proof of mirror symmetry for non-convex FJRW theory

Abstract

We provide a mirror symmetry theorem in a range of cases where the state-of-the-art techniques relying on concavity or convexity do not apply. More specifically, we work on a family of FJRW potentials named after Fan, Jarvis, Ruan, and Witten's quantum singularity theory and viewed as the counterpart of a non-convex Gromov--Witten potential via the physical LG/CY correspondence. The main result provides an explicit formula for Polishchuk and Vaintrob's virtual cycle in genus zero. In the non-concave case of the so-called chain invertible polynomials, it yields a compatibility theorem with the FJRW virtual cycle and a proof of mirror symmetry for FJRW theory.