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Bridgeland stability condition

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nonabelian homological algebra

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Idea

Bridgeland stability conditions on a triangulated category are certain data which give a derived analogue of the Mumford’s stability.

Definition

Let 𝒜\mathcal{A} be an abelian category and K(𝒜)K(\mathcal{A}) be its Grothendieck group. A stability function, sometimes also called a central charge, is a group homomorphism Z:K(𝒜)Z: K(\mathcal{A})\to \mathbb{C} such that for all non-zero objects, the image of ZZ lies in the semi-upper half plane H={rexp(iπϕ):rH=\{r exp(i\pi \phi) : r > 0,00, 0 < ϕ1}\phi \leq 1\}. The phase of an object is just the ϕ\phi that occurs in the representation from HH. Alternatively, by plotting Z(E)Z(E) in the complex plane the phase is the argument (slope) divided by π\pi. The phase of EE will be denoted ϕ(E)\phi(E).

An object EE is called semi-stable if for all non-trivial subobjects FEF\subset E we have the property that ϕ(F)ϕ(E)\phi(F)\leq \phi(E). An object EE is called stable if for all non-trivial, proper subojects FEF\subset E we have the property that ϕ(F)\phi(F) < ϕ(E)\phi(E).

A stability function Z:K(𝒜)Z:K(\mathcal{A})\to \mathbb{C} is said to have the Harder-Narasimhan property if for any non-zero object EE there exists a finite filtration by subobjects 0=E 0E 1E n=E0=E_0 \subset E_1 \subset \cdots \subset E_n =E such that the quotients F i=E i/E i1F_i=E_i/E_{i-1} are all semi-stable and satisfy ϕ(F 1)\phi(F_1) > ϕ(F 2)\phi(F_2) > \cdots > ϕ(F n)\phi(F_n).

Suppose 𝒟\mathcal{D} is a triangulated category (usually arising as the derived category of some abelian category). A slicing, 𝒫\mathcal{P}, is a choice of full additive subcategories 𝒫(ϕ)\mathcal{P}(\phi) for each ϕ\phi \in \mathbb{R} satisfying

  1. 𝒫(ϕ+1)=𝒫(ϕ)[1]\mathcal{P}(\phi +1)=\mathcal{P}(\phi)[1]
  2. If ϕ 1\phi_1 < ϕ 2\phi_2 and A j𝒫(ϕ j)A_j\in \mathcal{P}(\phi_j), then Hom(A 1,A 2)=0Hom(A_1, A_2)=0.
  3. Any object has a finite filtration by the slicing: If E𝒟E\in \mathcal{D}, then there exists ϕ 1\phi_1 > \cdots > ϕ n\phi_n and a sequence 0=E 0E 1E n=E0=E_0\to E_1 \to \cdots \to E_n =E such that the cone E j1E jF jE j1[1]E_{j-1}\to E_j \to F_j \to E_{j-1}[1] satisfies F j𝒫(ϕ j)F_j\in \mathcal{P}(\phi_j).

A stability condition on 𝒟\mathcal{D} is a pair σ=(Z,𝒫)\sigma = (Z, \mathcal{P}) consisting of a stability function and slicing satisfying the relation that given a non-zero object E𝒫(ϕ)E\in \mathcal{P}(\phi), then there is a non-zero positive real number m(E)m(E) such that Z(E)=m(E)exp(iπϕ)Z(E)=m(E)exp(i\pi \phi). This justifies the repeated notation of ϕ\phi, since this says that if an object lies in a particular slice ϕ\phi, then it must also have phase ϕ\phi.

Key Results

Bridgeland proves that an equivalent way to give a stability is to specify a heart of a bounded tt-structure on 𝒟\mathcal{D} and give a stability function the heart that satisfies the Harder-Narasimhan property.

Under reasonable hypotheses, one can put a natural topology on the space of stability conditions, Stab(𝒟)Stab(\mathcal{D}), under which the space becomes a complex manifold. Most work using this fact has been done where 𝒟=D b(Coh(X))\mathcal{D}=D^b(Coh(X)) where XX is a smooth, projective variety over \mathbb{C} so that 𝒟\mathcal{D} is \mathbb{C}-linear and K(𝔻)K(\mathbb{D}) is finitely generated.

Stab(X)Stab(X) has a locally finite wall and chamber decomposition. The philosophy is that if one fixes a numerical condition vv and considers the moduli space of σ\sigma-stable sheaves as σ\sigma varies through Stab(X)Stab(X), then the moduli spaces M σ(v)M σ(v)M_\sigma(v)\simeq M_{\sigma '}(v) should be isomorphic if σ\sigma and σ\sigma' are in the same chamber. Bayer and Macri have shown this is true for K3 surfaces, and upon crossing a wall the moduli spaces are related by a birational map.

Examples

A motivating example is the following. Let XX be a non-singular, projective curve over \mathbb{C}. Let 𝒜=Coh(X)\mathcal{A}=Coh(X) be the category of coherent sheaves on XX. The standard stability function is Z(E)=deg(E)+irk(E)Z(E)=-deg(E) + i rk(E). The classical notion of the slope of a vector bundle is μ(E)=rk(E)deg(E)\mu(E)=\frac{rk(E)}{deg(E)}. When constructing a moduli of vector bundles using GIT one needs to consider only slope (semi-)stable vector bundles. One can immediately see that a vector bundle is slope (semi-)stable if and only if it is (semi-)stable with respect to this stability function. Thus stability conditions are a generalization of classical notions of stability.

References

Bridgeland work came out as a way to formalize ideas on Π\Pi-stability in physics works.

  • M. R. Douglas, D-branes, categories and N=1N=1 supersymmetry, J.Math.Phys. 42 (2001) 2818–2843;Dirichlet branes, homological mirror symmetry, and stability_, Proc. ICM, Vol. III (Beijing, 2002), 395–408, Higher Ed. Press, Beijing, 2002

General

Introductions and lectures

Relation to stable branes in string theory

Relation to moduli space theory

  • Arend Bayer, Emaneule Macri, Projectivity and Birational Geometry of Bridgeland Moduli Spaces (arXiv:1203.4613)

Revised on November 19, 2013 03:35:42 by Zoran Škoda (161.53.130.104)