6-dimensional FJRW theories of the simple-elliptic singularities

TL;DR

This paper explicitly computes genus zero potentials for 6-dimensional FJRW theories related to simple-elliptic singularities and establishes their CY/LG correspondence with Gromov-Witten theory of a specific orbifold, including explicit transformations.

Contribution

It provides explicit formulas for the genus zero potentials and constructs the CY/LG correspondence for these FJRW theories with detailed variable transformations.

Findings

Explicit genus zero potentials for three FJRW theories of $ ilde E_7$

Construction of the CY/LG correspondence via Givental's group actions

Reconstruction of potentials up to a scaling using FJRW axioms

Abstract

We give explicitly in the closed formulae the genus zero primary potentials of the three 6-dimensional FJRW theories of the simple-elliptic singularity $\tilde E_7$ with the non-maximal symmetry groups. For each of these FJRW theories we establish the CY/LG correspondence to the Gromov-Witten theory of the orbifold $[\mathcal{E}/ (\mathbb{Z}/2\mathbb{Z})]$ --- the orbifold quotient of the elliptic curve by the hyperelliptic involution. Namely, we give explicitly the Givental's group elements, whose actions on the partition function of the Gromov--Witten theory of $[\mathcal{E}/ (\mathbb{Z}/2\mathbb{Z})]$ give up to a linear change of the variables the partition functions of the FJRW theories mentioned. We keep track of the linear changes of the variables needed. We show that using only the axioms of Fan--Jarvis--Ruan, the genus zero potential can only be reconstructed up to a scaling.