Derived equivalences of gentle algebras via Fukaya categories

TL;DR

This paper establishes a geometric framework linking gentle algebras to Fukaya categories, classifies their symmetries, and provides criteria for derived equivalences, leading to new classifications and connections to stacky curves.

Contribution

It introduces a geometric model for gentle algebras via Fukaya categories and classifies their symmetries to determine derived equivalences.

Findings

A geometric model associating gentle algebras with surfaces and line fields.

A classification of the action of the mapping class group on line fields.

A criterion for derived equivalence based on numerical invariants.

Abstract

Following the approach of Haiden-Katzarkov-Kontsevich arXiv:1409.8611, to any homologically smooth graded gentle algebra $A$ we associate a triple $(\Sigma_A, \Lambda_A; \eta_A)$, where $\Sigma_A$ is an oriented smooth surface with non-empty boundary, $\Lambda_A$ is a set of stops on $\partial \Sigma_A$ and $\eta_A$ is a line field on $\Sigma_A$, such that the derived category of perfect dg-modules of $A$ is equivalent to the partially wrapped Fukaya category of $(\Sigma_A, \Lambda_A ;\eta_A)$. Modifying arguments of Johnson and Kawazumi, we classify the orbit decomposition of the action of the (symplectic) mapping class group of $\Sigma_A$ on the homotopy classes of line fields. As a result we obtain a sufficient criterion for homologically smooth graded gentle algebras to be derived equivalent. Our criterion uses numerical invariants generalizing those given by Avella-Alaminos-Geiss in math/0607348, as well as some other numerical invariants. As an application, we find many new cases when the AAG-invariants determine the derived Morita class. As another application, we establish some derived equivalences between the stacky nodal curves considered in arXiv:1705.06023.