Autoequivalences of Fukaya categories of surfaces and graded gentle algebras

Abstract

We compute the derived Picard groups of partially wrapped Fukaya categories of surfaces in the sense of Haiden-Katzarkov-Kontsevich and the related graded gentle algebras. This includes the wrapped cases as introduced by Bocklandt. An important ingredient for our proof in characteristic zero is the exponential map from Hochschild cohomology to the derived Picard group introduced in recent work by the author. In positive characteristics, we combine deformation theory and formality results for Hochschild complexes to prove our results. Along the way we show that the surface together with its decorations forms a complete derived invariant of partially wrapped Fukaya categories and we prove analogous results for graded gentle algebras. This removes all previous restrictions from earlier results of this kind.