Classification of derived-equivalent algebras via formal dissections of the associated orbifold surface

Establish that for any graded skew-gentle algebra A and its associated graded orbifold surface with stops S = (S, E, n), every graded associative algebra B that is perfect derived equivalent to A is realized as the cohomology algebra H'(AA) of the A∞ category AA constructed from a formal dissection A of S. Equivalently, demonstrate that the graded associative algebras arising from formal dissections of S form a class closed under derived equivalence and exhaust all graded associative algebras derived equivalent to A.

Background

The paper develops a comprehensive framework for partially wrapped Fukaya categories of graded orbifold surfaces with stops and shows these categories admit formal generators whose cohomology algebras are graded skew-gentle algebras (Theorem 1.6, Theorems 8.5–8.9). Building on this, the authors conjecture a classification principle: any graded associative algebra derived equivalent to a given graded skew-gentle algebra should arise from a formal dissection of the associated orbifold surface, thereby closing the class under derived equivalence.

This conjecture extends the well-known closure under derived equivalence for ungraded gentle algebras (Schröer–Zimmermann) and aligns with recent progress on derived invariants via line fields for gentle algebras. The conjecture aims to characterize the entire derived equivalence class of a graded skew-gentle algebra geometrically through formal dissections of the corresponding orbifold surface.