$6$-dimensional FJRW theories of the simple–elliptic singularities

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 Ẽ7 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 elliptic orbifold [E/(Z/2Z)] -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 [E/(Z/2Z)] 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.