Virasoro constraints in quantum singularity theories

TL;DR

This paper introduces Virasoro operators for Landau-Ginzburg pairs and conjectures their annihilation of the FJRW theory's total ancestor potential, proving it in specific cases and exploring links to mirror symmetry and Calabi-Yau correspondence.

Contribution

It proposes a new Virasoro constraint framework for FJRW theory of Landau-Ginzburg pairs and proves it in several important cases.

Findings

Virasoro operators are introduced for Landau-Ginzburg pairs.

The conjecture that these operators annihilate the total ancestor potential is supported in multiple cases.

Connections among Virasoro constraints, mirror symmetry, and Landau-Ginzburg/Calabi-Yau correspondence are discussed.

Abstract

We introduce Virasoro operators for any Landau-Ginzburg pair (W, G) where W is a non-degenerate quasi-homogeneous polynomial and G is a certain group of diagonal symmetries. We propose a conjecture that the total ancestor potential of the FJRW theory of the pair (W,G) is annihilated by these Virasoro operators. We prove the conjecture in various cases, including: (1) invertible polynomials with the maximal group, (2) some two-variable polynomials with the minimal group, (3) certain Calabi-Yau polynomials with groups. We also discuss the connections among Virasoro constraints, mirror symmetry of Landau-Ginzburg models, and Landau-Ginzburg/Calabi-Yau correspondence.