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 derived 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 formula_1-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 formula_2 be a triangulated category. A slicing formula_3 of formula_2 is a collection of full additive subcategories formula_5 for each formula_6 such that
The last property should be viewed as axiomatically imposing the existence of Harder–Narasimhan filtrations on elements of the category formula_2.
A Bridgeland stability condition on a triangulated category formula_2 is a pair formula_20 consisting of a slicing formula_3 and a group homomorphism formula_22, where formula_23 is the Grothendieck group of formula_2, called a central charge, satisfying
It is convention to assume the category formula_2 is essentially small, so that the collection of all stability conditions on formula_2 forms a set formula_30. In good circumstances, for example when formula_31 is the derived category of coherent sheaves on a complex manifold formula_32, 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 formula_33 on the category formula_2 and a central charge formula_35 on the heart formula_36 of this t-structure which satisfies the Harder–Narasimhan property above. An element formula_37 is semi-stable (resp. stable) with respect to the stability condition formula_20 if for every surjection formula_39 for formula_40, we have formula_41 where formula_42 and similarly for formula_43.