# Bridgeland stability condition

In mathematics, and especially algebraic geometry, a Bridgeland stability condition, defined by Tom Bridgeland, is an algebro-geometric stability condition defined on elements of a triangulated category. The case of original interest and particular importance is when this triangulated category is the derived category of coherent sheaves on a Calabi–Yau manifold, and this situation has fundamental links to string theory and the study of D-branes.
Such stability conditions were introduced in a rudimentary form by Michael Douglas called $\Pi$-stability and used to study BPS B-branes in string theory. This concept was made precise by Bridgeland, who phrased these stability conditions categorically, and initiated their study mathematically.

## Definition

The definitions in this section are presented as in the original paper of Bridgeland, for arbitrary triangulated categories. Let ${\mathcal {D}}$ be a triangulated category.

### Slicing of triangulated categories

A slicing ${\mathcal {P}}$ of ${\mathcal {D}}$ is a collection of full additive subcategories ${\mathcal {P}}(\varphi )$ for each $\varphi \in \mathbb {R}$ such that
* ${\mathcal {P}}(\varphi )[1]={\mathcal {P}}(\varphi +1)$ for all $\varphi$, where $[1]$ is the shift functor on the triangulated category,
* if $\varphi _{1}>\varphi _{2}$ and $A\in {\mathcal {P}}(\varphi _{1})$ and $B\in {\mathcal {P}}(\varphi _{2})$, then $\operatorname {Hom} (A,B)=0$, and
* for every object $E\in {\mathcal {D}}$ there exists a finite sequence of real numbers $\varphi _{1}>\varphi _{2}>\cdots >\varphi _{n}$ and a collection of triangles
    with $A_{i}\in {\mathcal {P}}(\varphi _{i})$ for all $i$.
The last property should be viewed as axiomatically imposing the existence of Harder–Narasimhan filtrations on elements of the category ${\mathcal {D}}$.

### Stability conditions

A Bridgeland stability condition on a triangulated category ${\mathcal {D}}$ is a pair $(Z,{\mathcal {P}})$ consisting of a slicing ${\mathcal {P}}$ and a group homomorphism $Z:K({\mathcal {D}})\to \mathbb {C}$, where $K({\mathcal {D}})$ is the Grothendieck group of ${\mathcal {D}}$, called a central charge, satisfying
* if $0\neq E\in {\mathcal {P}}(\varphi )$ then $Z(E)=m(E)\exp(i\pi \varphi )$ for some strictly positive real number $m(E)\in \mathbb {R} _{>0}$.
It is convention to assume the category ${\mathcal {D}}$ is essentially small, so that the collection of all stability conditions on ${\mathcal {D}}$ forms a set $\operatorname {Stab} ({\mathcal {D}})$. In good circumstances, for example when ${\mathcal {D}}={\mathcal {D}}^{b}\operatorname {Coh} (X)$ is the derived category of coherent sheaves on a complex manifold $X$, this set actually has the structure of a complex manifold itself.

#### Technical remarks about stability condition

It is shown by Bridgeland that the data of a Bridgeland stability condition is equivalent to specifying a bounded t-structure ${\mathcal {P}}(>0)$ on the category ${\mathcal {D}}$ and a central charge $Z:K({\mathcal {A}})\to \mathbb {C}$ on the heart ${\mathcal {A}}={\mathcal {P}}((0,1])$ of this t-structure which satisfies the Harder–Narasimhan property above.
An element $E\in {\mathcal {A}}$ is semi-stable (resp. stable) with respect to the stability condition $(Z,{\mathcal {P}})$ if for every surjection $E\to F$ for $F\in {\mathcal {A}}$, we have $\varphi (E)\leq ({\text{resp.}}<)\,\varphi (F)$ where $Z(E)=m(E)\exp(i\pi \varphi (E))$ and similarly for $F$.

## Examples
### From the Harder–Narasimhan filtration

Recall the Harder–Narasimhan filtration for a smooth projective curve $X$ implies for any coherent sheaf $E$ there is a filtration
   $0=E_{0}\subset E_{1}\subset \cdots \subset E_{n}=E$
such that the factors $E_{j}/E_{j-1}$ have slope $\mu _{i}={\text{deg}}/{\text{rank}}$. Set $\phi (E)=\arg(-\mathrm {deg} (E)+i\mathrm {rank} (E))$. We can extend this filtration to a bounded complex of sheaves $E^{\bullet }$ by considering the filtration on the cohomology sheaves $E^{i}=H^{i}(E^{\bullet })[+i]$. From this observation, the pair $(Z,{\mathcal {P}})$ is a Bridgeland stability condition if the central charge and the slicing are defined to be $Z(E)=-\mathrm {deg} (E)+\mathrm {rank} (E),{\mathcal {P}}(\phi )=\{E^{\bullet }\colon \mu {\text{-semistable of phase }}\phi \}.$

### Elliptic curves

There is an analysis by Bridgeland for the case of Elliptic curves. He finds there is an equivalence
   ${\text{Stab}}(X)/{\text{Aut}}(X)\cong {\text{GL}}^{+}(2,\mathbb {R} )/{\text{SL}}(2,\mathbb {Z} )$
where ${\text{Stab}}(X)$ is the set of stability conditions and ${\text{Aut}}(X)$ is the set of autoequivalences of the derived category $D^{b}(X)$.