Todor Milanov, Yefeng Shen
A simple elliptic singularity of type $E_N^{(1,1)}$ ($N=6,7,8$) can be described in terms of a marginal deformation of an invertible polynomial $W$. In the papers \cite{KS} and \cite{MR} the authors proved mirror symmetry and used it to prove quasi-modularity of certain elliptic orbifold-$\mathbb{P}^1$s. The choice of the polynomial $W$ and its marginal deformation is not unique however. In this paper, we investigate the mirror symmetry phenomenon for all possible choices of $W$ and particular choices of the marginal deformation. Our main result is that any special limit point from Saito-Givental theory of invertible simple elliptic singularities is mirror to either Gromov-Witten theory of elliptic orbifold $\mathbb{P}^1$ or FJRW theory of an invertible simple elliptic singularity with diagonal symmetries. In each case the mirror symmetry is governed by a certain system of hypergeometric equations. They are classified by the Milnor number of the singularity and the $j$-invariant at the special limit point. In particular, by evaluating the corresponding monodromy groups we prove that the modular groups of the elliptic orbifold $\mathbb{P}^1$s studied in \cite{KS,MR} are $\Gamma(3)$, $\Gamma(4)$, and $\Gamma(6).$
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http://arxiv.org/abs/1210.6862
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