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

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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

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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

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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

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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

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From the Harder–Narasimhan filtration

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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

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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

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  1. ^ Douglas, M.R., Fiol, B. and Römelsberger, C., 2005. Stability and BPS branes. Journal of High Energy Physics, 2005(09), p. 006.
  2. ^ a b c d 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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