Contractible stability spaces and faithful braid group actions

Abstract

We prove that any “finite-type” component of a stability space of a triangulated category is contractible. The motivating example of such a component is the stability space of the Calabi–Yau–$N$ category $\mathcal{D}\left({\mathrm{\Gamma }}_{N}Q\right)$ associated to an ADE Dynkin quiver. In addition to showing that this is contractible we prove that the braid group $Br\left(Q\right)$ acts freely upon it by spherical twists, in particular that the spherical twist group $Br\left({\mathrm{\Gamma }}_{N}Q\right)$ is isomorphic to $Br\left(Q\right)$. This generalises the result of Brav–Thomas for the $N=2$ case. Other classes of triangulated categories with finite-type components in their stability spaces include locally finite triangulated categories with finite-rank Grothendieck group and discrete derived categories of finite global dimension.