Contractible stability spaces and faithful braid group actions

Qiu, Yu¹ ; Woolf, Jon²

¹ Yau Mathematical Sciences Center, Tsinghua University, Beijing, China
² Department of Mathematical Sciences, University of Liverpool, Liverpool, United Kingdom

Geometry & topology, Tome 22 (2018) no. 6, pp. 3701-3760.

Résumé

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 $\mathcal{D}\left({\Gamma }_{N}Q\right)$ 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 $Br\left({\Gamma }_{N}Q\right)$ 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.

Détail

DOI : 10.2140/gt.2018.22.3701

Classification : 18E30, 20F36, 13F60, 32Q55
Keywords: stability conditions, Calabi–Yau categories, spherical twists, braid groups

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