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
The definitions in this section are presented as in the original paper of Bridgeland, for arbitrary triangulated categories. Let
l{D}
A slicing
l{P}
l{D}
l{P}(\varphi)
\varphi\inR
l{P}(\varphi)[1]=l{P}(\varphi+1)
\varphi
[1]
\varphi1>\varphi2
A\inl{P}(\varphi1)
B\inl{P}(\varphi2)
\operatorname{Hom}(A,B)=0
E\inl{D}
\varphi1>\varphi2> … >\varphin
with
Ai\inl{P}(\varphii)
i
The last property should be viewed as axiomatically imposing the existence of Harder–Narasimhan filtrations on elements of the category
l{D}
A Bridgeland stability condition on a triangulated category
l{D}
(Z,l{P})
l{P}
Z:K(l{D})\toC
K(l{D})
l{D}
0\neE\inl{P}(\varphi)
Z(E)=m(E)\exp(i\pi\varphi)
m(E)\inR>
It is convention to assume the category
l{D}
l{D}
\operatorname{Stab}(l{D})
l{D}=l{D}b\operatorname{Coh}(X)
X
l{P}(>0)
l{D}
Z:K(l{A})\toC
l{A}=l{P}((0,1])
An element
E\inl{A}
(Z,l{P})
E\toF
F\inl{A}
\varphi(E)\le(resp.<)\varphi(F)
Z(E)=m(E)\exp(i\pi\varphi(E))
F
Recall the Harder–Narasimhan filtration for a smooth projective curve
X
E
such that the factors0=E0\subsetE1\subset … \subsetEn=E
Ej/Ej-1
\mui=deg/rank
E\bullet
Ei=Hi(E\bullet)[+i]
i | |
E | |
j |
=\mui+j
for the central charge.\phi:K(X)\toR
There is an analysis by Bridgeland for the case of Elliptic curves. He finds[3] there is an equivalence
whereStab(X)/Aut(X)\congGL+(2,R)/SL(2,Z)
Stab(X)
Aut(X)
Db(X)
An