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.

[1] This concept was made precise by Bridgeland, who phrased these stability conditions categorically, and initiated their study mathematically.

[2] The definitions in this section are presented as in the original paper of Bridgeland, for arbitrary triangulated categories.

is a collection of full additive subcategories

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

A Bridgeland stability condition on a triangulated category

consisting of a slicing

, 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

of this t-structure which satisfies the Harder–Narasimhan property above.

stable) with respect to the stability condition

) exp ⁡ ( i π φ (

Recall the Harder–Narasimhan filtration for a smooth projective curve

implies for any coherent sheaf

We can extend this filtration to a bounded complex of sheaves

by considering the filtration on the cohomology sheaves

and defining the slope of

, giving a function

There is an analysis by Bridgeland for the case of Elliptic curves.

{\displaystyle {\text{Stab}}(X)/{\text{Aut}}(X)\cong {\text{GL}}^{+}(2,\mathbb {R} )/{\text{SL}}(2,\mathbb {Z} )}

is the set of stability conditions and

is the set of autoequivalences of the derived category