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]

Definition

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

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.[2]

An element is semi-stable (resp. 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 . 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

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

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

References

  1. Douglas, M.R., Fiol, B. and Römelsberger, C., 2005. Stability and BPS branes. Journal of High Energy Physics, 2005(09), p. 006.
  2. Bridgeland, Tom (2006-02-08). "Stability conditions on triangulated categories". arXiv:math/0212237.
  3. Uehara, Hokuto (2015-11-18). "Autoequivalences of derived categories of elliptic surfaces with non-zero Kodaira dimension". pp. 10–12. arXiv:1501.06657 [math.AG].

Papers

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