Twisted Jacobian algebras as endomorphism algebras of equivariant matrix factorizations

TL;DR

This paper establishes an isomorphism between twisted Jacobian rings and endomorphism rings of equivariant matrix factorizations, providing new tools for studying singularities and mirror symmetry.

Contribution

It introduces the concept of twisted Jacobian algebras and proves their isomorphism to endomorphism algebras of equivariant matrix factorizations, linking singularity invariants to categorical structures.

Findings

Twisted Jacobian rings are isomorphic to endomorphism rings of twisted diagonal matrix factorizations.

The results offer a new approach to studying Floer theory of Lagrangian submanifolds.

Application to homological mirror symmetry suggests broader implications.

Abstract

Given a polynomial $W$ with an isolated singularity, we can consider the Jacobian ring as an invariant of the singularity. If in addition we have a group action on the polynomial ring with $W$ fixed, we are led to consider the twisted Jacobian ring which reflects the equivariant structure as well. Our main result is to show that the twisted Jacobian ring is isomorphic to an endomorphism ring of the "twisted diagonal" matrix factorization. As an application, we suggest a way to investigate Floer theory of Lagrangian submanifolds which represent homological mirror functors.