and
nonabelian homological algebra
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 $A$ 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 $A$ is a “twisted direct sum” of $B$ and $C$ (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 $A$ 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 $\mathcal{A}$ be an abelian category and $K(\mathcal{A})$ be its Grothendieck group. A stability function, sometimes also called a central charge, is a group homomorphism $Z: K(\mathcal{A})\to \mathbb{C}$ such that for all non-zero objects, the image of $Z$ lies in the semi-upper half plane $H=\{r exp(i\pi \phi) : r$ > $0, 0$ < $\phi \leq 1\}$. The phase of an object is just the $\phi$ that occurs in the representation from $H$. Alternatively, by plotting $Z(E)$ in the complex plane the phase is the argument (slope) divided by $\pi$. The phase of $E$ will be denoted $\phi(E)$.
An object $E$ is called semi-stable if for all non-trivial subobjects $F\subset E$ we have the property that $\phi(F)\leq \phi(E)$. An object $E$ is called stable if for all non-trivial, proper subojects $F\subset E$ we have the property that $\phi(F)$ < $\phi(E)$.
A stability function $Z:K(\mathcal{A})\to \mathbb{C}$ is said to have the Harder-Narasimhan property if for any non-zero object $E$ there exists a finite filtration by subobjects $0=E_0 \subset E_1 \subset \cdots \subset E_n =E$ such that the quotients $F_i=E_i/E_{i-1}$ are all semi-stable and satisfy $\phi(F_1)$ > $\phi(F_2)$ > $\cdots$ > $\phi(F_n)$.
Suppose $\mathcal{D}$ is a triangulated category (usually arising as the derived category of some abelian category). A slicing, $\mathcal{P}$, is a choice of full additive subcategories $\mathcal{P}(\phi)$ for each $\phi \in \mathbb{R}$ satisfying
A stability condition on $\mathcal{D}$ is a pair $\sigma = (Z, \mathcal{P})$ consisting of a stability function and slicing satisfying the relation that given a non-zero object $E\in \mathcal{P}(\phi)$, then there is a non-zero positive real number $m(E)$ such that $Z(E)=m(E)exp(i\pi \phi)$. This justifies the repeated notation of $\phi$, since this says that if an object lies in a particular slice $\phi$, then it must also have phase $\phi$.
Bridgeland proves that an equivalent way to give a stability is to specify a heart of a bounded $t$-structure on $\mathcal{D}$ 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, $Stab(\mathcal{D})$, under which the space becomes a complex manifold. Most work using this fact has been done where $\mathcal{D}=D^b(Coh(X))$ where $X$ is a smooth, projective variety over $\mathbb{C}$ so that $\mathcal{D}$ is $\mathbb{C}$-linear and $K(\mathcal{D})$ is finitely generated.
$Stab(X)$ has a locally finite wall and chamber decomposition. The philosophy is that if one fixes a numerical condition $v$ and considers the moduli space of $\sigma$-stable sheaves as $\sigma$ varies through $Stab(X)$, then the moduli spaces $M_\sigma(v)\simeq M_{\sigma '}(v)$ should be isomorphic if $\sigma$ and $\sigma'$ 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 $X$ be a non-singular, projective curve over $\mathbb{C}$. Let $\mathcal{A}=Coh(X)$ be the category of coherent sheaves on $X$. The standard stability function is $Z(E)=-deg(E) + i rk(E)$. The classical notion of the slope of a vector bundle is $\mu(E)=\frac{rk(E)}{deg(E)}$. 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.
Bridgeland’s work was motivated as a formalizatioon of ideas on $\Pi$-stability of D-branes for the topological string, as discussed in
Michael Douglas, D-branes, categories and $N=1$ 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
Paul Aspinwall, D-Branes on Calabi-Yau Manifolds (arXiv:hep-th/0403166)
Tom Bridgeland, Spaces of stability conditions, Proc. of symposia in pure math. 80, 2009, math/0611510.
bourwiki: Bridgeland stability conditions
R. Pandharipande, R.P. Thomas, Stable pairs and BPS invariants, arXiv:0711.3899
Maxim Kontsevich, Yan Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435
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: