Contractible stability spaces and faithful braid group actions

TL;DR

This paper proves that certain components of stability spaces in triangulated categories are contractible and that braid groups act freely on them, generalizing previous results and providing new insights into their structure.

Contribution

It establishes the contractibility of finite-type components of stability spaces and demonstrates free braid group actions, extending known results to broader classes of categories.

Findings

Finite-type stability space components are contractible.

Braid groups act freely via spherical twists on these components.

The spherical twist group is isomorphic to the braid group of the quiver.

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 $\mathcal{D}(\Gamma_N Q)$ associated to an ADE Dynkin quiver. In addition to showing that this is contractible we prove that the braid group $\operatorname{Br}(Q)$ acts freely upon it by spherical twists, in particular that the spherical twist group $\operatorname{Br}(\Gamma_N Q)$ is isomorphic to $\operatorname{Br}(Q)$. This generalises Brav-Thomas' result 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.