Английская Википедия: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 <math>\Pi</math>-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 <math>\mathcal{D}</math> be a triangulated category.
Slicing of triangulated categories
A slicing <math>\mathcal{P}</math> of <math>\mathcal{D}</math> is a collection of full additive subcategories <math>\mathcal{P}(\varphi)</math> for each <math>\varphi\in \mathbb{R}</math> such that
- <math>\mathcal{P}(\varphi)[1] = \mathcal{P}(\varphi+1)</math> for all <math>\varphi</math>, where <math>[1]</math> is the shift functor on the triangulated category,
- if <math>\varphi_1 > \varphi_2</math> and <math>A\in \mathcal{P}(\varphi_1)</math> and <math>B\in \mathcal{P}(\varphi_2)</math>, then <math>\operatorname{Hom}(A,B)=0</math>, and
- for every object <math>E\in \mathcal{D}</math> there exists a finite sequence of real numbers <math>\varphi_1>\varphi_2>\cdots>\varphi_n</math> and a collection of triangles
- with <math>A_i\in \mathcal{P}(\varphi_i)</math> for all <math>i</math>.
The last property should be viewed as axiomatically imposing the existence of Harder–Narasimhan filtrations on elements of the category <math>\mathcal{D}</math>.
Stability conditions
A Bridgeland stability condition on a triangulated category <math>\mathcal{D}</math> is a pair <math>(Z,\mathcal{P})</math> consisting of a slicing <math>\mathcal{P}</math> and a group homomorphism <math>Z: K(\mathcal{D}) \to \mathbb{C}</math>, where <math>K(\mathcal{D})</math> is the Grothendieck group of <math>\mathcal{D}</math>, called a central charge, satisfying
- if <math>0\ne E\in \mathcal{P}(\varphi)</math> then <math>Z(E) = m(E) \exp(i\pi \varphi)</math> for some strictly positive real number <math>m(E) \in \mathbb{R}_{> 0}</math>.
It is convention to assume the category <math>\mathcal{D}</math> is essentially small, so that the collection of all stability conditions on <math>\mathcal{D}</math> forms a set <math>\operatorname{Stab}(\mathcal{D})</math>. In good circumstances, for example when <math>\mathcal{D} = \mathcal{D}^b \operatorname{Coh}(X)</math> is the derived category of coherent sheaves on a complex manifold <math>X</math>, 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 <math>\mathcal{P}(>0)</math> on the category <math>\mathcal{D}</math> and a central charge <math>Z: K(\mathcal{A})\to \mathbb{C}</math> on the heart <math>\mathcal{A} = \mathcal{P}((0,1])</math> of this t-structure which satisfies the Harder–Narasimhan property above.[2]
An element <math>E\in\mathcal{A}</math> is semi-stable (resp. stable) with respect to the stability condition <math>(Z,\mathcal{P})</math> if for every surjection <math>E \to F</math> for <math>F\in \mathcal{A}</math>, we have <math>\varphi(E) \le (\text{resp.}<) \, \varphi(F)</math> where <math>Z(E) = m(E) \exp(i\pi \varphi(E))</math> and similarly for <math>F</math>.
Examples
From the Harder–Narasimhan filtration
Recall the Harder–Narasimhan filtration for a smooth projective curve <math>X</math> implies for any coherent sheaf <math>E</math> there is a filtration
<math>0 = E_0 \subset E_1 \subset \cdots \subset E_n = E</math>
such that the factors <math>E_j/E_{j-1}</math> have slope <math>\mu_i=\text{deg}/\text{rank}</math>. We can extend this filtration to a bounded complex of sheaves <math>E^\bullet</math> by considering the filtration on the cohomology sheaves <math>E^i = H^i(E^\bullet)[+i]</math> and defining the slope of <math>E^i_j = \mu_i + j</math>, giving a function
<math>\phi : K(X) \to \mathbb{R}</math>
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
<math>\text{Stab}(X)/\text{Aut}(X) \cong \text{GL}^+(2,\mathbb{R})/\text{SL}(2,\mathbb{Z})</math>
where <math>\text{Stab}(X)</math> is the set of stability conditions and <math>\text{Aut}(X)</math> is the set of autoequivalences of the derived category <math>D^b(X)</math>.
References
Papers
- Stability conditions on <math>A_n</math> singularities
- Interactions between autoequivalences, stability conditions, and moduli problems
- ↑ Douglas, M.R., Fiol, B. and Römelsberger, C., 2005. Stability and BPS branes. Journal of High Energy Physics, 2005(09), p. 006.
- ↑ 2,0 2,1 2,2 2,3 Шаблон:Cite arXiv
- ↑ Шаблон:Cite arXiv
