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

Todor Milanov (Kavli Institute for the Physics and Mathematics of the Universe (WPI), Todai Institutes for Advanced Study, University of Tokyo, Kashiwa, Chiba, Japan)

Yefeng Shen (Kavli Institute for the Physics and Mathematics of the Universe (WPI), Todai Institutes for Advanced Study, University of Tokyo, Kashiwa, Chiba, Japan)

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 $\Gamma(6)$.

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