nLab Bridgeland stability condition

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

Theorems

Stable homotopy theory

stable homotopy theory

Contents

Idea

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

Definition

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

1. $\mathcal{P}(\phi +1)=\mathcal{P}(\phi)[1]$
2. If $\phi_1$ < $\phi_2$ and $A_j\in \mathcal{P}(\phi_j)$, then $Hom(A_1, A_2)=0$.
3. Any object has a finite filtration by the slicing: If $E\in \mathcal{D}$, then there exists $\phi_1$ > $\cdots$ > $\phi_n$ 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 $F_j\in \mathcal{P}(\phi_j)$.

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 $\phi$.

Key Results

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.

Examples

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.

References

Bridgeland work came out as a way to formalize ideas on $\Pi$-stability in physics works.

• M. R. Douglas, D-branes, categories and $N=1$ supersymmetry, J.Math.Phys. 42 (2001) 2818–2843;Dirichlet branes, homological mirror symmetry, and stability_, Proc. ICM, Vol. III (Beijing, 2002), 395–408, Higher Ed. Press, Beijing, 2002

Relation to stable branes in string theory

Relation to stable B-branes:

Relation to moduli space theory

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

Revised on May 29, 2015 08:13:59 by Tim Kirschner? (132.252.63.190)