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.^[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 ${\mathcal {D}}$ 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.^[2]

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

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

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

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

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

References

^ Douglas, M.R., Fiol, B. and Römelsberger, C., 2005. Stability and BPS branes. Journal of High Energy Physics, 2005(09), p. 006.
^ ^a ^b ^c ^d Bridgeland, Tom (2006-02-08). "Stability conditions on triangulated categories". arXiv:math/0212237.
^ Uehara, Hokuto (2015-11-18). "Autoequivalences of derived categories of elliptic surfaces with non-zero Kodaira dimension". pp. 10–12. arXiv:1501.06657 [math.AG].

Papers

[douglas-1] Douglas, M.R., Fiol, B. and Römelsberger, C., 2005. Stability and BPS branes. Journal of High Energy Physics, 2005(09), p. 006.

[bridgeland-2] Bridgeland, Tom (2006-02-08). "Stability conditions on triangulated categories". arXiv:math/0212237.

[3] Uehara, Hokuto (2015-11-18). "Autoequivalences of derived categories of elliptic surfaces with non-zero Kodaira dimension". pp. 10–12. arXiv:1501.06657 [math.AG].

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