nLab
Bridgeland stability condition

Idea

Bridgeland stability conditions on a triangulated category are certain data which give a derived analogue of the Mumford’s stability.

Definition

Let $𝒜$ be an abelian category and $K(𝒜)$ be its Grothendieck group. A stability function, sometimes also called a central charge, is a group homomorphism $Z:K(𝒜)\to ℂ$ such that for all non-zero objects, the image of $Z$ lies in the semi-upper half plane $H=\{r\exp (i\pi \varphi ):r$ > $0,0$ < $\varphi \le 1\}$. The phase of an object is just the $\varphi$ 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 $\varphi (E)$.

An object $E$ is called semi-stable if for all non-trivial subobjects $F\subset E$ we have the property that $\varphi (F)\le \varphi (E)$. An object $E$ is called stable if for all non-trivial, proper subojects $F\subset E$ we have the property that $\varphi (F)$ < $\varphi (E)$.

A stability function $Z:K(𝒜)\to ℂ$ 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 $\varphi (F_1)$ > $\varphi (F_2)$ > $\cdots$ > $\varphi (F_n)$.

Suppose $𝒟$ is a triangulated category (usually arising as the derived category of some abelian category). A slicing, $𝒫$, is a choice of full additive subcategories $𝒫(\varphi )$ for each $\varphi \in ℝ$ satisfying

1. $𝒫(\varphi +1)=𝒫(\varphi )[1]$
2. If ${\varphi }_1$ < ${\varphi }_2$ and $A_j\in 𝒫({\varphi }_j)$, then $\mathrm{Hom}(A_1,A_2)=0$.
3. Any object has a finite filtration by the slicing: If $E\in 𝒟$, then there exists ${\varphi }_1$ > $\cdots$ > ${\varphi }_n$ and a sequence $0=E_0\to E_1\to \cdots \to E_n=E$ such that the cone $E_{j-1}\to E_j\to F_j\to E_{j-1}[1]$ satisfies $F_j\in 𝒫({\varphi }_j)$.

A stability condition on $𝒟$ is a pair $\sigma =(Z,𝒫)$ consisting of a stability function and slicing satisfying the relation that given a non-zero object $E\in 𝒫(\varphi )$, then there is a non-zero positive real number $m(E)$ such that $Z(E)=m(E)\exp (i\pi \varphi )$. This justifies the repeated notation of $\varphi$, since this says that if an object lies in a particular slice $\varphi$, then it must also have phase $\varphi$.

Key Results

Bridgeland proves that an equivalent way to give a stability is to specify a heart of a bounded $t$-structure on $𝒟$ 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, $\mathrm{Stab}(𝒟)$, under which the space becomes a complex manifold. Most work using this fact has been done where $𝒟=D^b(\mathrm{Coh}(X))$ where $X$ is a smooth, projective variety over $ℂ$ so that $𝒟$ is $ℂ$-linear and $K(𝔻)$ is finitely generated.

$\mathrm{Stab}(X)$ 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 $\mathrm{Stab}(X)$, then the moduli spaces $M_{\sigma }(v)\simeq M_{\sigma \prime }(v)$ should be isomorphic if $\sigma$ and $\sigma \prime$ 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.

Examples

A motivating example is the following. Let $X$ be a non-singular, projective curve over $ℂ$. Let $𝒜=\mathrm{Coh}(X)$ be the category of coherent sheaves on $X$. The standard stability function is $Z(E)=-\mathrm{deg}(E)+i\mathrm{rk}(E)$. The classical notion of the slope of a vector bundle is $\mu (E)=\frac{\mathrm{rk}(E)}{\mathrm{deg}(E)}$. 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.

Related concepts

• wall crossing of BPS states

References

General

• Tom Bridgeland, Stability conditions on triangulated categories, Ann. of Math. 166 (2007) 317–345,math.AG/0212237;

• 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

Introductions and lectures

• Anton Geraschenko, notes from an introductory talk (pdf)

Relation to stable branes in string theory

• Aaron Bergman?, Stability Conditions and Branes at Singularities (arXiv:hep-th/0702092)

Relation to moduli space theory

• Arend Bayer, Emaneule Macri, Projectivity and Birational Geometry of Bridgeland Moduli Spaces (arXiv:1203.4613)