Contractibility of spaces of stability conditions

Abstract:

To each triangulated category T one can associate a complex manifold Stab(T) of stability conditions on T. The expectation is that Stab(T) is always contractible, and this has been verified in many examples. In this talk, I will explain how to prove that any `finite-type’ component of Stab(T) is contractible, by considering a cellular stratification of Stab(T) arising from the decomposition into regions in which the heart of the stability condition remains constant. I will use examples from the theory of quivers to illustrate the results, and, I hope, explain all the technical terms above. If there is time I will also discuss why the braid and spherical twist groups of a Dynkin quiver are isomorphic. This is joint work with Yu Qiu.