Contractible stability spaces and faithful braid group actions

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 $D (Γ_{N} Q)$ associated to an ADE Dynkin quiver. In addition to showing that this is contractible we prove that the braid group $Br (Q)$ acts freely upon it by spherical twists, in particular that the spherical twist group $Br (Γ_{N} Q)$ is isomorphic to $Br (Q)$ . 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.

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Keywords

stability conditions, Calabi–Yau categories, spherical twists, braid groups

Mathematical Subject Classification 2010

Primary: 18E30, 20F36

Secondary: 13F60, 32Q55

References

Publication

Received: 7 November 2017

Revised: 8 February 2018

Accepted: 13 March 2018

Published: 23 September 2018

Proposed: Richard Thomas

Seconded: Jim Bryan, Frances Kirwan

Authors

Yu Qiu
Yau Mathematical Sciences Center Tsinghua University Beijing, China

Jon Woolf
Department of Mathematical Sciences University of Liverpool Liverpool United Kingdom

Volume 22, issue 6 (2018)

Yu Qiu and Jon Woolf

Abstract