The Period map for quantum cohomology of P2\mathbb{P}^2

We invert the period map defined by the second structure connection of quantum cohomology of $\mathbb{P}^2$. For small quantum cohomology the inverse is given explicitly in terms of the Eisenstein series $E_4$ and $E_6$, while for big quantum cohomology the inverse is determined perturbatively as a Taylor series expansion whose coefficients are quasi-modular forms.Comment: 58 pages. Typos fixed. The previous version was extended in the following way: we proved the Laplace transform version of the $\Gamma$-conjecture for $\mathbb{P}^2$, worked out the structure of a Schwarz triangular map, and added an appendix explaining the relation to vertex algebras and W-constraints. Version accepted in Adv. in Mat

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 $\mathbb{P}^1_{3,3,3}$, $\mathbb{P}^1_{4,4,2}$, 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 $\Gamma(6)$.Comment: 28 pages. arXiv admin note: substantial text overlap with arXiv:1210.686