The space of stability structures, Stab(C), of a triangulated category C was introduced in seminal work of Bridgeland  based on a proposal of M.R. Douglas on D-branes in super-conformal field theories. The data of a stability structure, σ , is a distinguished class of objects in C, the semistable ones, each with a phase φ ∈ R, and an additive DOI 10.1007/s10240-017-0095-y 248 F. HAIDEN, L. KATZARKOV, M. KONTSEVICH map Z : K0(C) → C, the central charge. These have to satisfy a number of axioms, see Section 5.1. Although the definition seems somewhat strange at first from a mathematical point of view, it leads to the remarkable fact, proven by Bridgeland, that Stab(C) naturally has the structure of a complex manifold, possibly infinite-dimensional. Furthermore, Stab(C) comes with an action of Aut(C) and the universal cover of GL+(2,R). Thanks to the efforts of a number of people, the structure of Stab(C) is understood in many particular cases, see e.g
In this paper we give a construction of phantom categories, i.e. admissible triangulated subcategories in bounded derived categories of coherent sheaves on smooth projective varieties that have trivial Hochschild homology and trivial Grothendieck group. We also prove that these phantom categories are phantoms in a stronger sense and all their higher K-groups are trivial too.