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

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 ${\mathbb{P}}_{3,3,3}^1$, ${\mathbb{P}}_{4,4,2}^1$, 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 $\Gamma (3)$, 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 $\Gamma (6)$.

28 pages. arXiv admin note: substantial text overlap with arXiv:1210.6862

References in corpus (4)

• FJRW rings and Landau-Ginzburg Mirror Symmetry
• Gromov-Witten theory of elliptic orbifold P^1 and quasi-modular forms
• Landau-Ginzburg/Calabi-Yau Correspondence of all Genera for Elliptic Orbifold $p^1$
• Gromov-Witten theory and cycle-valued modular forms

Cited by in corpus (1)

• Ramanujan Identities and Quasi-Modularity in Gromov-Witten Theory