Bridgeland stability conditions on a triangulated category are certain data which give a derived analogue of the Mumford’s stability.
The concept originally arose as a formalization of reaction processes of D-branes for the B-model topological string. These are (hypothetical) physical objects that appear in different species labeled by objects in a triangulated derived category (of quasicoherent sheaves on some Calabi-Yau variety). Whenever there is a process
by which a brane of type may decay into a brane of type B and a brane of type C (much like a chemical reaction), this is witnessed by the fact that there is a homotopy fiber sequence (a distinguished triangle in the triangulated category) of the form
(See Aspinwall 04, search the document for the keyword “decay”.)
Mathematically such a homotopy fiber sequence in a triangulated category precisely expresses the fact that is a “twisted direct sum” of and (extension, semidirect product), hence much like the plain direct sum, but with some “interaction” included.
In addition there are then Bridgeland stability conditions on such triangulated categories of topological D-branes, which determine which of these reaction processes lead to stable compounds (whence the term!), i.e. whether, in the above example, the brane of type will really decay into branes of type B and C, or if conversely the latter will fuse. (See again Aspinwall 04, search the document for the keyword “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.
Michael Douglas, D-branes, categories and supersymmetry, J.Math.Phys. 42 (2001) 2818–2843;
Michael Douglas Dirichlet branes, homological mirror symmetry, and stability, Proc. ICM, Vol. III (Beijing, 2002), 395–408, Higher Ed. Press, Beijing, 2002
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
Relation to stable B-branes: