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 CalabiâÂÂYau 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.
The definitions in this section are presented as in the original paper of Bridgeland, for arbitrary triangulated categories. Let be a triangulated category.
A slicing of is a collection of full additive subcategories for each such that
The last property should be viewed as axiomatically imposing the existence of HarderâÂÂNarasimhan filtrations on elements of the category .
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
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.
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 HarderâÂÂNarasimhan property above.
An element is semi-stable (resp. stable) with respect to the stability condition if for every surjection for , we have where and similarly for .
Recall the HarderâÂÂNarasimhan filtration for a smooth projective curve implies for any coherent sheaf there is a filtration<blockquote></blockquote>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
There is an analysis by Bridgeland for the case of Elliptic curves. He finds there is an equivalence<blockquote></blockquote>where is the set of stability conditions and is the set of autoequivalences of the derived category .