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

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 ${\mathbb{P}}_{3,3,3}^1$, ${\mathbb{P}}_{4,4,2}^1$, and ${\mathbb{P}}_{6,3,2}^1$ are quasi-modular forms on an appropriate modular group. While the modular group for ${\mathbb{P}}_{3,3,3}^1$ is $\mathrm{\Gamma }(3)$, 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 $\mathrm{\Gamma }(4)$ and $\mathrm{\Gamma }(6)$.