Asian Journal of Mathematics

Volume 26 (2022)

Number 1

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

Pages: 45 – 80

DOI: https://dx.doi.org/10.4310/AJM.2022.v26.n1.a3

Author

Alexey Basalaev (Faculty of Mathematics, National Research University Higher School of Economics, Moscow, Russian Federation; and Skolkovo Institute of Science and Technology, Moscow, Russian Federation)

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 elliptic 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.

Keywords

FJRW theories, Landau–Ginzburg models, mirror symmetry

2010 Mathematics Subject Classification

Primary 14N35, 53D45. Secondary 14J33.

Received 9 January 2018

Accepted 1 September 2021

Published 30 January 2023