Abstract

Following the approach of Haiden-Katzarkov-Kontsevich [15], to any homologically smooth $\mathbb{Z}$ -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 [7], 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 [21].