On auto-equivalences and complete derived invariants of gentle algebras

Abstract

We study triangulated categories which can be modeled by an oriented marked surface $\mathcal{S}$ and a line field $\eta$ on $\mathcal{S}$. This includes bounded derived categories of gentle algebras and -- conjecturally -- all partially wrapped Fukaya categories introduced by Haiden-Katzarkov-Kontsevich. We show that triangle equivalences between such categories induce diffeomorphisms of the associated surfaces preserving orientation, marked points and line fields up to homotopy. This shows that the pair $(\mathcal{S}, \eta)$ is a triangle invariant of such categories and prove that it is a complete derived invariant for gentle algebras of arbitrary global dimension. We deduce that the group of auto-equivalences of a gentle algebra is an extension of the stabilizer subgroup of $\eta$ in the mapping class group and a group, which we describe explicitely in case of triangular gentle algebras. We show further that diffeomorphisms associated to spherical twists are Dehn twists.