Algebraic stability conditions and contractible stability spaces

Suppose that C is either a locally-finite triangulated category with finite rank Grothendieck group, or a discrete derived category of finite global dimension. We prove that any component of the space of stability conditions on C is contractible (and that there is only one component in the discrete case). More generally, we prove that any ‘finite-type’ component of a stability space is contractible. In particular, the principal component of the stability space associated to the Calabi–Yau-N Ginzburg algebra of an ADE Dynkin quiver is contractible. These results generalise and unify various known ones for stability spaces of specific categories, and settle some conjectures about the stability spaces associated to Dynkin quivers, and to their Calabi–Yau-N Ginzburg algebras.

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