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Bridgeland stability condition

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Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Stable homotopy theory

Contents

Idea

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

The concept originally arose as a formalization of reaction processes of D-branes for the B-model topological string. These are (hypothetical) physical objects that appear in different species labeled by objects in a triangulated derived category (of quasicoherent sheaves on some Calabi-Yau variety). Whenever there is a process

ABC A \leftrightarrow B \oplus C

by which a brane of type AA may decay into a brane of type B and a brane of type C (much like a chemical reaction), this is witnessed by the fact that there is a homotopy fiber sequence (a distinguished triangle in the triangulated category) of the form

BAC. B \longrightarrow A \longrightarrow C \,.

(See Aspinwall 04, search the document for the keyword “decay”.)

Mathematically such a homotopy fiber sequence in a triangulated category precisely expresses the fact that AA is a “twisted direct sum” of BB and CC (extension, semidirect product), hence much like the plain direct sum, but with some “interaction” included.

In addition there are then Bridgeland stability conditions on such triangulated categories of topological D-branes, which determine which of these reaction processes lead to stable compounds (whence the term!), i.e. whether, in the above example, the brane of type AA will really decay into branes of type B and C, or if conversely the latter will fuse. (See again Aspinwall 04, search the document for the keyword “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(\mathcal{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’s work was motivated as a formalizatioon of ideas on Π\Pi-stability of D-branes for the topological string, as discussed in

  • Michael Douglas, D-branes, categories and N=1N=1 supersymmetry, J.Math.Phys. 42 (2001) 2818–2843;

  • Michael Douglas Dirichlet branes, homological mirror symmetry, and stability, Proc. ICM, Vol. III (Beijing, 2002), 395–408, Higher Ed. Press, Beijing, 2002

  • Paul Aspinwall, D-Branes on Calabi-Yau Manifolds (arXiv:hep-th/0403166)

General

Introductions and lectures

Relation to stable branes in string theory

Relation to stable B-branes:

Relation to moduli space theory

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

Revised on December 14, 2016 04:02:28 by Urs Schreiber (46.183.103.17)