Closure of graded gentle algebras under derived equivalence

Establish that the class of graded gentle algebras is closed under derived equivalence. Equivalently, show that for any graded smooth surface with stops S = (S, E, n), every formal generator of the partially wrapped Fukaya category W(S) is obtained from a formal dissection of S.

Background

For smooth surfaces, partially wrapped Fukaya categories admit descriptions via graded gentle algebras and formal generators coming from dissections. In the ungraded setting, closure under derived equivalence for gentle algebras is known (Schröer–Zimmermann), and recent work has provided derived invariants using line fields.

The authors posit the graded analogue: graded gentle algebras should remain within their class under derived equivalence, and geometrically, any formal generator of W(S) for a graded smooth surface should be realizable by a formal dissection, mirroring the orbifold case where formal dissections classify derived-equivalent algebras.