Global mirror symmetry for invertible simple elliptic singularities

Abstract

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 a mirror symmetry statement for some particular choices of $W$ and used it to prove quasi-modularity of Gromov-Witten invariants for certain elliptic orbifold ${\mathbb{P}}^1$s. However, the choice of the polynomial $W$ and its marginal deformation ${\varphi }_{\mu }$ are not unique. In this paper, we investigate the global mirror symmetry phenomenon for the one-parameter family $W+\sigma {\varphi }_{\mu }$. In each case the mirror symmetry is governed by a certain system of hypergeometric equations. We conjecture that the Saito-Givental theory of $W+\sigma {\varphi }_{\mu }$ at any special limit $\sigma$ is mirror to either the Gromov-Witten theory of an elliptic orbifold ${\mathbb{P}}^1$ or the Fan-Jarvis-Ruan-Witten theory of an invertible simple elliptic singularity with diagonal symmetries, and the limits are classified by the Milnor number of the singularity and the $j$-invariant at the special limit. We prove the conjecture when $W$ is a Fermat polynomial. We also prove that the conjecture is true at the Gepner point $\sigma =0$ in all other cases.

The last section of version 1 is removed, because we added more details to it and it became a separate paper. 47 pages

References in corpus (4)

• Landau-Ginzburg/Calabi-Yau correspondence, global mirror symmetry and Orlov equivalence
• Mirror Symmetry and Integral Variations of Hodge Structure Underlying One Parameter Families of Calabi-Yau Threefolds
• Quantum BCOV theory on Calabi-Yau manifolds and the higher genus B-model
• FJRW-Rings and Landau-Ginzburg Mirror Symmetry in Two Dimensions

Cited by in corpus (4)

• Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for P^1_{a,b,c}
• A Mirror Theorem between Landau-Ginzburg models
• On the L2-Hodge theory of Landau-Ginzburg models
• On quantum cohomology ring of elliptic ${\mathbb{P}}^1$ orbifolds