Green groupoids of 2-Calabi--Yau categories, derived Picard actions, and hyperplane arrangements

Abstract

We present a construction of (faithful) group actions via derived equivalences in the general categorical setting of algebraic 2-Calabi--Yau triangulated categories. To each algebraic 2-Calabi--Yau category $\mathscr{C}$ satisfying standard mild assumptions, we associate a groupoid $\mathscr{G}_{ \mathscr{C} }$, named the green groupoid of $\mathscr{C}$, defined in an intrinsic homological way. Its objects are given by a set of representatives $\operatorname{mrig} \mathscr{C}$ of the equivalence classes of basic maximal rigid objects of $\mathscr{C}$, arrows are given by mutation, and relations are given by equating monotone (green) paths in the silting order. In this generality we construct a homomorphsim from the green groupoid $\mathscr{G}_{ \mathscr{C} }$ to the derived Picard groupoid of the collection of endomorphism rings of representatives of $\operatorname{mrig} \mathscr{C}$ in a Frobenius model of $\mathscr{C}$; the latter canonically acts by triangle equivalences between the derived categories of the rings. We prove that the constructed representation of the green groupoid $\mathscr{G}_{ \mathscr{C} }$ is faithful if the index chamber decompositions of the split Grothendieck groups of basic maximal rigid objects of $\mathscr{C}$ come from hyperplane arrangements. If $\Sigma^2 \cong \operatorname{id}$ and $\mathscr{C}$ has finitely many equivalence classes of basic maximal rigid objects, we prove that $\mathscr{G}_{ \mathscr{C} }$ is isomorphic to a Deligne groupoid of a hyperplane arrangement and that the representation of this groupoid is faithful.