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 ${\displaystyle \mu _{i}={\text{deg}}/{\text{rank}}}$ . Set ${\displaystyle \phi (E)=\arg(-\mathrm {deg} (E)+i\mathrm {rank} (E))}$ . We can extend this filtration to a bounded complex of sheaves ${\displaystyle E^{\bullet }}$  by considering the filtration on the cohomology sheaves ${\displaystyle E^{i}=H^{i}(E^{\bullet })[+i]}$ . From this observation, the pair ${\displaystyle (Z,{\mathcal {P}})}$  is a Bridgeland stability condition if the central charge and the slicing are defined to be $${\displaystyle Z(E)=-\mathrm {deg} (E)+\mathrm {rank} (E),{\mathcal {P}}(\phi )=\{E^{\bullet }\colon \mu {\text{-semistable of phase }}\phi \}.}$$ 

Elliptic curves

There is an analysis by Bridgeland for the case of Elliptic curves. He finds^[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

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

Papers

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