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

TL;DR

This paper proves a homological mirror symmetry equivalence between categories associated with two-variable invertible polynomials and their Berglund-H"ubsch transposes, extending previous results to a broader class of singularities.

Contribution

It establishes a new homological mirror symmetry result for two-variable invertible polynomials, using explicit tilting objects and Lefschetz thimbles, generalizing prior cases.

Findings

Category of matrix factorizations is quasi-equivalent to Fukaya-Seidel category

Constructs explicit tilting object on the B-side

Compares with basis of Lefschetz thimbles on the A-side

Abstract

Given a two-variable invertible polynomial, we show that its category of maximally-graded matrix factorisations is quasi-equivalent to the Fukaya-Seidel category of its Berglund-H\"ubsch transpose. This was previously shown for Brieskorn-Pham and $D$-type singularities by Futaki-Ueda. The proof involves explicit construction of a tilting object on the B-side, and comparison with a specific basis of Lefschetz thimbles on the A-side.