Matrix factorizations via Koszul duality

TL;DR

This paper develops a curved Koszul duality framework for graded curved coalgebras and algebras, providing new tools to analyze matrix factorizations and their homological properties.

Contribution

It introduces a curved Koszul duality theory and a homological perturbation lemma applicable to curved (co)algebras, linking Hochschild homology with matrix factorizations.

Findings

Established curved Koszul duality for graded curved coalgebras and algebras.

Connected Hochschild homology of curved (co)algebras with matrix factorizations.

Extended results to orbifold and graded cases.

Abstract

In this paper we prove a version of curved Koszul duality for Z/2Z-graded curved coalgebras and their coBar differential graded algebras. A curved version of the homological perturbation lemma is also obtained as a useful technical tool for studying curved (co)algebras and precomplexes. The results of Koszul duality can be applied to study the category of matrix factorizations MF(R,W). We show how Dyckerhoff's generating results fit into the framework of curved Koszul duality theory. This enables us to clarify the relationship between the Borel-Moore Hochschild homology of curved (co)algebras and the ordinary Hochschild homology of the category MF(R,W). Similar results are also obtained in the orbifold case and in the graded case.