Computational Techniques in FJRW Theory with Applications to Landau-Ginzburg Mirror Symmetry

TL;DR

This paper develops computational methods to determine the FJRW theory's A-model structure, enabling the explicit calculation of Frobenius manifolds for complex singularities and non-maximal symmetry groups, advancing Landau-Ginzburg mirror symmetry.

Contribution

It introduces new techniques for computing FJRW A-model structures, including cases with non-maximal symmetry groups, and explicitly determines Frobenius manifolds for 27 previously unknown singularities.

Findings

Computed Frobenius manifolds for 27 singularities

Determined A-model structures in non-maximal symmetry cases

Enhanced understanding of Landau-Ginzburg mirror symmetry

Abstract

The Landau-Ginzburg A-model, given by FJRW theory, defines a cohomological field theory, but in most examples is very difficult to compute, especially when the symmetry group is not maximal. We give some methods for finding the A-model structure. In many cases our methods completely determine the previously unknown A-model Frobenius manifold structure. In the case where these Frobenius manifolds are semisimple, this can be shown to determine the structure of the higher genus potential as well. We compute the Frobenius manifold structure for 27 of the previously unknown unimodal and bimodal singularities and corresponding groups, including 13 cases using a non-maximal symmetry group.