The modular group for the total ancestor potential of Fermat simple elliptic singularities

TL;DR

This paper determines the specific modular groups associated with the orbifold Gromov-Witten invariants of certain elliptic orbifold projective lines, completing the classification of their modular properties.

Contribution

It proves that the modular groups for the orbifold projective lines $ ext{P}^1_{4,4,2}$ and $ ext{P}^1_{6,3,2}$ are $ ext{Gamma}(4)$ and $ ext{Gamma}(6)$, respectively, extending previous results.

Findings

Identified the modular group for $ ext{P}^1_{4,4,2}$ as $ ext{Gamma}(4)$.

Identified the modular group for $ ext{P}^1_{6,3,2}$ as $ ext{Gamma}(6)$.

Confirmed the quasi-modularity of orbifold Gromov-Witten invariants for these cases.

Abstract

In a series of papers \cite{KS,MR}, Krawitz, Milanov, Ruan, and Shen have verified the so-called Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence for simple elliptic singularities $E_N^{(1,1)}$ ($N=6,7,8$). As a byproduct it was also proved that the orbifold Gromov--Witten invariants of the orbifold projective lines $\mathbb{P}^1_{3,3,3}$, $\mathbb{P}^1_{4,4,2}$, and $\mathbb{P}^1_{6,3,2}$ are quasi-modular forms on an appropriate modular group. While the modular group for $\mathbb{P}^1_{3,3,3}$ is $\Gamma(3)$, the modular groups in the other two cases were left unknown. The goal of this paper is to prove that the modular groups in the remaining two cases are respectively $\Gamma(4)$ and $\Gamma(6)$.