# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1903.01351