Bridgeland stability condition explained

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

\Pi

-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. Let

l{D}

be a triangulated category.

Slicing of triangulated categories

A slicing

l{P}

of

l{D}

is a collection of full additive subcategories

l{P}(\varphi)

for each

\varphi\inR

such that

l{P}(\varphi)[1]=l{P}(\varphi+1)

for all

\varphi

, where

[1]

is the shift functor on the triangulated category,

\varphi1>\varphi2

and

A\inl{P}(\varphi1)

and

B\inl{P}(\varphi2)

, then

\operatorname{Hom}(A,B)=0

, and

E\inl{D}

there exists a finite sequence of real numbers

\varphi1>\varphi2>>\varphin

and a collection of triangles

with

Ai\inl{P}(\varphii)

for all

i

.

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

l{D}

.

Stability conditions

A Bridgeland stability condition on a triangulated category

l{D}

is a pair

(Z,l{P})

consisting of a slicing

l{P}

and a group homomorphism

Z:K(l{D})\toC

, where

K(l{D})

is the Grothendieck group of

l{D}

, called a central charge, satisfying

0\neE\inl{P}(\varphi)

then

Z(E)=m(E)\exp(i\pi\varphi)

for some strictly positive real number

m(E)\inR>

.

It is convention to assume the category

l{D}

is essentially small, so that the collection of all stability conditions on

l{D}

forms a set

\operatorname{Stab}(l{D})

. In good circumstances, for example when

l{D}=l{D}b\operatorname{Coh}(X)

is the derived category of coherent sheaves on a complex manifold

X

, this set actually has the structure of a complex manifold itself.

Technical remarks about stability condition

l{P}(>0)

on the category

l{D}

and a central charge

Z:K(l{A})\toC

on the heart

l{A}=l{P}((0,1])

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

An element

E\inl{A}

is semi-stable (resp. stable) with respect to the stability condition

(Z,l{P})

if for every surjection

E\toF

for

F\inl{A}

, we have

\varphi(E)\le(resp.<)\varphi(F)

where

Z(E)=m(E)\exp(i\pi\varphi(E))

and similarly for

F

.

Examples

From the Harder–Narasimhan filtration

Recall the Harder–Narasimhan filtration for a smooth projective curve

X

implies for any coherent sheaf

E

there is a filtration

0=E0\subsetE1\subset\subsetEn=E

such that the factors

Ej/Ej-1

have slope

\mui=deg/rank

. We can extend this filtration to a bounded complex of sheaves

E\bullet

by considering the filtration on the cohomology sheaves

Ei=Hi(E\bullet)[+i]

and defining the slope of
i
E
j

=\mui+j

, giving a function

\phi:K(X)\toR

for the central charge.

Elliptic curves

There is an analysis by Bridgeland for the case of Elliptic curves. He finds[3] there is an equivalence

Stab(X)/Aut(X)\congGL+(2,R)/SL(2,Z)

where

Stab(X)

is the set of stability conditions and

Aut(X)

is the set of autoequivalences of the derived category

Db(X)

.

References

Papers

Notes and 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 . math/0212237 .
  3. Uehara . Hokuto . 2015-11-18 . Autoequivalences of derived categories of elliptic surfaces with non-zero Kodaira dimension . 1501.06657 . 10–12. math.AG .