  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.
