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 -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 be a triangulated category.

Slicing of triangulated categories

A slicing of is a collection of full additive subcategories for each such that

for all, where is the shift functor on the triangulated category,
if and and, then, and
for every object there exists a finite sequence of real numbers and a collection of triangles

The last property should be viewed as axiomatically imposing the existence of Harder–Narasimhan filtrations on elements of the category.

Stability conditions

A Bridgeland stability condition on a triangulated category is a pair consisting of a slicing and a group homomorphism, where is the Grothendieck group of, called a central charge, satisfying

if then for some strictly positive real number.

It is convention to assume the category is essentially small, so that the collection of all stability conditions on forms a set. In good circumstances, for example when is the derived category of coherent sheaves on a complex manifold, 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 on the category and a central charge on the heart of this t-structure which satisfies the Harder–Narasimhan property above.
An element is semi-stable with respect to the stability condition if for every surjection for, we have where and similarly for.

Examples

From the Harder–Narasimhan filtration

Recall the Harder–Narasimhan filtration for a smooth projective curve implies for any coherent sheaf there is a filtration

such that the factors have slope. Set. We can extend this filtration to a bounded complex of sheaves by considering the filtration on the cohomology sheaves. From this observation, the pair is a Bridgeland stability condition if the central charge and the slicing are defined to be

Elliptic curves

There is an analysis by Bridgeland for the case of Elliptic curves. He finds there is an equivalence

where is the set of stability conditions and is the set of autoequivalences of the derived category.

Papers

Category:Geometry
Category:String theory
Category:Algebraic geometry