Abstract
Following the approach of Haiden–Katzarkov–Kontsevich (Publ Math Inst Hautes Études Sci 126:247–318, 2017), 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 Avella et al. (J Pure Appl Algebra 212(1):228–243, 2008), 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 Lekili and Polishchuk (J Topology 11:615–444, 2018)
Highlights
Given a Liouville manifold (M, ω = dλ), a rigorous definition of the compact Fukaya category, F(M), appears in the monograph [27]
The orientations on each Lagrangian determine a Z2-grading on F(M), and the spin structures enter in orienting the moduli spaces of holomorphic polygons that enter into the definition of structure constants of the A∞ operations
We can think of line fields as trivializations of the circle fibration (1.1), in particular, the set G( ) has a natural structure of a torsor over the group of homotopy classes of maps → S1, i.e., with H 1( )
Summary
Note that from the numerical invariants of Theorem 1.8 one can recover the genus of the surface and the numbers of stops on the boundary components, so if these invariants match the corresponding partially wrapped Fukaya categories are equivalent. We use this result to construct derived equivalences between gentle algebras, introduced by Assem and Skowrónski in [3].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
