On homological mirror symmetry for chain type polynomials

TL;DR

This paper investigates homological mirror symmetry for chain polynomials, providing a proof on the B-side and a detailed sketch on the A-side, advancing understanding of categorical structures in mirror symmetry.

Contribution

It establishes a recursion of directed A-infinity categories for chain polynomials and proves it on the B-side, offering insights into the categorical aspects of mirror symmetry.

Findings

Proves the recursion of categories on the B-side

Provides a detailed sketch for the A-side argument

Advances categorical understanding in mirror symmetry

Abstract

We consider Takahashi's categorical interpretation of the Berglund-Hubsch mirror symmetry conjecture for invertible polynomials in the case of chain polynomials. Our strategy is based on a stronger claim that the relevant categories satisfy a recursion of directed $A_{\infty}$-categories, which may be of independent interest. We give a full proof of this claim on the B-side. On the A-side we give a detailed sketch of an argument, which falls short of a full proof because of certain missing foundational results in Fukaya-Seidel categories, most notably a generation statement.