Bridgeland stability conditions on a triangulated category are certain data which give a derived analogue of the Mumford’s stability.
Let be an abelian category and be its Grothendieck group. A stability function, sometimes also called a central charge, is a group homomorphism such that for all non-zero objects, the image of lies in the semi-upper half plane > < . The phase of an object is just the that occurs in the representation from . Alternatively, by plotting in the complex plane the phase is the argument (slope) divided by . The phase of will be denoted .
An object is called semi-stable if for all non-trivial subobjects we have the property that . An object is called stable if for all non-trivial, proper subojects we have the property that < .
A stability function is said to have the Harder-Narasimhan property if for any non-zero object there exists a finite filtration by subobjects such that the quotients are all semi-stable and satisfy > > > .
A stability condition on is a pair consisting of a stability function and slicing satisfying the relation that given a non-zero object , then there is a non-zero positive real number such that . This justifies the repeated notation of , since this says that if an object lies in a particular slice , then it must also have phase .
Bridgeland proves that an equivalent way to give a stability is to specify a heart of a bounded -structure on and give a stability function the heart that satisfies the Harder-Narasimhan property.
Under reasonable hypotheses, one can put a natural topology on the space of stability conditions, , under which the space becomes a complex manifold. Most work using this fact has been done where where is a smooth, projective variety over so that is -linear and is finitely generated.
has a locally finite wall and chamber decomposition. The philosophy is that if one fixes a numerical condition and considers the moduli space of -stable sheaves as varies through , then the moduli spaces should be isomorphic if and are in the same chamber. Bayer and Macri have shown this is true for K3 surfaces, and upon crossing a wall the moduli spaces are related by a birational map.
A motivating example is the following. Let be a non-singular, projective curve over . Let be the category of coherent sheaves on . The standard stability function is . The classical notion of the slope of a vector bundle is . When constructing a moduli of vector bundles using GIT one needs to consider only slope (semi-)stable vector bundles. One can immediately see that a vector bundle is slope (semi-)stable if and only if it is (semi-)stable with respect to this stability function. Thus stability conditions are a generalization of classical notions of stability.
bourwiki: Bridgeland stability conditions
R. Pandharipande, R.P. Thomas, Stable pairs and BPS invariants, arXiv:0711.3899
Rina Anno, Roman Bezrukavnikov, Ivan Mirković, A thin stringy moduli space for Slodowy slices, arxiv/1108.1563
Tom Bridgeland, Ivan Smith, Quadratic differentials as stability conditions, arxiv/1302.7030