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 ${\displaystyle \Pi }$-stability and used to study BPS B-branes in string theory.^[1] This concept was made precise by Bridgeland, who phrased these stability conditions categorically, and initiated their study mathematically.^[2]

Definition

The definitions in this section are presented as in the original paper of Bridgeland, for arbitrary triangulated categories.^[2] Let ${\displaystyle {\mathcal {D}}}$ be a triangulated category.

Slicing of triangulated categories

A slicing ${\displaystyle {\mathcal {P}}}$ of ${\displaystyle {\mathcal {D}}}$ is a collection of full additive subcategories ${\displaystyle {\mathcal {P}}(\varphi )}$ for each ${\displaystyle \varphi \in \mathbb {R} }$ such that

• ${\displaystyle {\mathcal {P}}(\varphi )[1]={\mathcal {P}}(\varphi +1)}$ for all ${\displaystyle \varphi }$, where ${\displaystyle [1]}$ is the shift functor on the triangulated category,
• if ${\displaystyle \varphi _{1}>\varphi _{2}}$ and ${\displaystyle A\in {\mathcal {P}}(\varphi _{1})}$ and ${\displaystyle B\in {\mathcal {P}}(\varphi _{2})}$, then ${\displaystyle \operatorname {Hom} (A,B)=0}$, and
• for every object ${\displaystyle E\in {\mathcal {D}}}$ there exists a finite sequence of real numbers ${\displaystyle \varphi _{1}>\varphi _{2}>\cdots >\varphi _{n}}$ and a collection of triangles

with ${\displaystyle A_{i}\in {\mathcal {P}}(\varphi _{i})}$ for all ${\displaystyle i}$.

The last property should be viewed as axiomatically imposing the existence of Harder–Narasimhan filtrations on elements of the category ${\displaystyle {\mathcal {D}}}$.

Stability conditions

A Bridgeland stability condition on a triangulated category ${\displaystyle {\mathcal {D}}}$ is a pair ${\displaystyle (Z,{\mathcal {P}})}$ consisting of a slicing ${\displaystyle {\mathcal {P}}}$ and a group homomorphism ${\displaystyle Z:K({\mathcal {D}})\to \mathbb {C} }$, where ${\displaystyle K({\mathcal {D}})}$ is the Grothendieck group of ${\displaystyle {\mathcal {D}}}$, called a central charge, satisfying

• if ${\displaystyle 0\neq E\in {\mathcal {P}}(\varphi )}$ then ${\displaystyle Z(E)=m(E)\exp(i\pi \varphi )}$ for some strictly positive real number ${\displaystyle m(E)\in \mathbb {R} _{>0}}$.

It is convention to assume the category ${\displaystyle {\mathcal {D}}}$ is essentially small, so that the collection of all stability conditions on ${\displaystyle {\mathcal {D}}}$ forms a set ${\displaystyle \operatorname {Stab} ({\mathcal {D}})}$. In good circumstances, for example when ${\displaystyle {\mathcal {D}}={\mathcal {D}}^{b}\operatorname {Coh} (X)}$ is the derived category of coherent sheaves on a complex manifold ${\displaystyle X}$, 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 ${\displaystyle {\mathcal {P}}(>0)}$ on the category ${\displaystyle {\mathcal {D}}}$ and a central charge ${\displaystyle Z:K({\mathcal {A}})\to \mathbb {C} }$ on the heart ${\displaystyle {\mathcal {A}}={\mathcal {P}}((0,1])}$ of this t-structure which satisfies the Harder–Narasimhan property above.^[2]

An element ${\displaystyle E\in {\mathcal {A}}}$ is semi-stable (resp. stable) with respect to the stability condition ${\displaystyle (Z,{\mathcal {P}})}$ if for every surjection ${\displaystyle E\to F}$ for ${\displaystyle F\in {\mathcal {A}}}$, we have ${\displaystyle \varphi (E)\leq ({\text{resp.}}<)\,\varphi (F)}$ where ${\displaystyle Z(E)=m(E)\exp(i\pi \varphi (E))}$ and similarly for ${\displaystyle F}$.

Examples

From the Harder–Narasimhan filtration

Recall the Harder–Narasimhan filtration for a smooth projective curve ${\displaystyle X}$ implies for any coherent sheaf ${\displaystyle E}$ there is a filtration

${\displaystyle 0=E_{0}\subset E_{1}\subset \cdots \subset E_{n}=E}$

such that the factors ${\displaystyle E_{j}/E_{j-1}}$ have slope ${\displaystyle \mu _{i}={\text{deg}}/{\text{rank}}}$. We can extend this filtration to a bounded complex of sheaves ${\displaystyle E^{\bullet }}$ by considering the filtration on the cohomology sheaves ${\displaystyle E^{i}=H^{i}(E^{\bullet })[+i]}$ and defining the slope of ${\displaystyle E_{j}^{i}=\mu _{i}+j}$, giving a function

${\displaystyle \phi :K(X)\to \mathbb {R} }$

for the central charge.

Elliptic curves

There is an analysis by Bridgeland for the case of Elliptic curves. He finds^[2][3] there is an equivalence

${\displaystyle {\text{Stab}}(X)/{\text{Aut}}(X)\cong {\text{GL}}^{+}(2,\mathbb {R} )/{\text{SL}}(2,\mathbb {Z} )}$

where ${\displaystyle {\text{Stab}}(X)}$ is the set of stability conditions and ${\displaystyle {\text{Aut}}(X)}$ is the set of autoequivalences of the derived category ${\displaystyle D^{b}(X)}$.

References

Papers

• Stability conditions on ${\displaystyle A_{n}}$ singularities
• Interactions between autoequivalences, stability conditions, and moduli problems