Homological Berglund-H\"ubsch-Henningson mirror symmetry for curve singularities

TL;DR

This paper proves a version of homological mirror symmetry for curve singularities, incorporating equivariance, and confirms a conjecture relating superpotentials and crepant resolutions, with implications for Landau-Ginzburg models.

Contribution

It establishes homological Berglund--Hübsch mirror symmetry for curve singularities with equivariance and proves a conjecture connecting superpotentials to crepant resolutions.

Findings

Proved homological Berglund--Hübsch mirror symmetry for curve singularities.

Confirmed the conjecture of Futaki and Ueda on equivariance and superpotentials.

Identified a tilting object in the B-model category of matrix factorizations.

Abstract

In this article, we establish homological Berglund--H\"ubsch mirror symmetry for curve singularities where the A--model incorporates equivariance, otherwise known as homological Berglund--H\"ubsch--Henningson mirror symmetry, including for certain deformations of categories. More precisely, we prove a conjecture of Futaki and Ueda in arXiv:1004.0078 which posits that the equivariance in the A-model can be incorporated by pulling back the superpotential to the total space of the corresponding crepant resolution. Along the way, we show that the B--model category of matrix factorisations has a tilting object whose length is the dimension of the state space of the FJRW A--model, a result which might be of independent interest for its implications in the Landau--Ginzburg analogue of Dubrovin's conjecture.