Contents Online

# Communications in Number Theory and Physics

## Volume 8 (2014)

### Number 2

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

Pages: 329 – 368

DOI: http://dx.doi.org/10.4310/CNTP.2014.v8.n2.a4

#### Authors

#### Abstract

In a series of papers [12, 15], Krawitz, Milanov, Ruan and Shen have verified the so-called Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence for simple elliptic singularities $E^{(1,1)}_N \; (N=6,7,8)$. As a by-product 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 $\Gamma(6)$.