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Bridgeland stability condition

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Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

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Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Stable homotopy theory

Contents

Idea

General

A stability condition (Bridgeland 02) on a triangulated category (hence on a stable model category/stable (∞,1)-category) 𝒟\mathcal{D} (Def. below) singles out some of its objects, to be called the “stable objects” (Def. below), that behave, to some extent, like indecomposable objects (whence “stable”), in that a version of Schur's lemma applies to them (Prop. below).

Indeed, in the degenerate case that 𝒟\mathcal{D} is the derived category of a semisimple category, the stable objects are just the irreducible objects.

In the case where 𝒟\mathcal{D} is the bounded derived category of coherent sheaves on some complex manifold, the general notion of stability reduces to the classical notion of slope-stability of coherent sheaves (“μ\mu-stability”, see Example below).

In other examples Bridgeland stability reproduces the derived analogue of David Mumford‘s concept of stability in geometric invariant theory (King 94).

There are in general different stability conditions on one and the same triangulated category, in fact their moduli space Stab(𝒟)Stab(\mathcal{D}) forms a complex manifold (Prop. below). The collection of stable objects in 𝒟\mathcal{D} in general depends on the choice of stability conditions, hence of the choice of point in this moduli space: There are in general codimension-1 submanifolds of Stab(𝒟)Stab(\mathcal{D}), called walls, such that the set of stable objects is constant close to both sides of these walls, but changes as the wall is crossed. This phenomenon is hence known as wall crossing.

The purely mathematical motivation for these definitions is, to a large extent, just their intrinsic richness. The concept finds its meaning in the concept of stability of D-branes in string theory and was abstracted from informal considerations about B-branes of the B-model topological string due to (Douglas-Fiol-Romerlsberger 00) and followups (“Π\Pi-stability”).

This string-theoretic interpretation also sheds light on the older notion of slope-stability of coherent sheaves, we discuss this below.

As stability of BPS D-branes

We explain here how to understand the mathematical formulation of stability conditions as a formalization of the concept of stable BPS states of D-branes, hence of extremal black branes. This yields a very simple and beautiful picture, which may be hard to extract from the original proposal due to Douglas-Fiol-Romerlsberger 00 (as witnessed by various reviews of the concept, see for instance Stellari 15, slide 9).

A key point to notice is that coherent sheaves and with them most of the kinds of objects in triangulated categories of relevance here, may be thought of as models for D-branes carrying D-brane charge.

Specifically, for XX a space, to be thought of as spacetime, a coherent sheaf EE over XX may be thought of as an abelian sheaf of sections of a kind of complex vector bundle over XX, whose fiber dimension is allowed to jump in some controlled way. Hence coherent sheaves are a slight generalization of complex vector bundles.

When considering D-branes on XX, such coherent sheaves/vector bundles EE appear as the Chan-Paton gauge fields on the D-brane.

objectEAAAAD-brane carrying mass and charge \text{object}\, E \phantom{AA}\leftrightarrow\phantom{AA} \text{D-brane carrying mass and charge}

From this perspective, the rank

rank(E) rank(E) \;\in\; \mathbb{R}

of EE is the number of coincident D-branes (there may be fractional D-branes, and, once we pass to the derived category, there may be anti D-branes, so that this number need not be a positive integer).

The D-branes have a fixed tension and hence a fixed mass-density, so that the total mass of the D-branes corresponding to EE is proportional to this rank. Ignoring the constant proportionality factor, we make this explicit by re-writing the rank as

M(E)rank(E). M(E) \;\coloneqq\; rank(E) \,.

In addition to their mass, the D-branes carry charge, called RR-charge. This is a generalization of the classical magnetic charge known from Dirac charge quantization. As explained there, magnetic charge reflected in a complex line bundle, as sourced by magnetic monopoles (D0-branes) is measured by the first Chern class c 1(E)c_1(E). In generalization of this, the total D-brane charge reflected in a Chan-Paton gauge field EE is proportional its Chern character ch(E)ch(E).

Q(E). Q(E) \;\in\; \mathbb{R} \,.

For coherent sheaves this is essentially what is classically called the degree of a coherent sheaf. Interpreting this as the charge carried by the D-brane, we may write

Q(E)=degree(E) Q(E) \;=\; degree(E)

Accordingly, D-branes EE have a charge density proportional to

ChargeDensity(E)Q(E)M(E)=degree(E)rank(E)=slope(E). ChargeDensity(E) \;\coloneqq\; \frac{ Q(E) }{ M(E) } \;=\; \frac{ degree(E) }{ rank(E) } \;=\; slope(E) \,.

This quotient is what is classically called the slope, as shown on the right. The terminology comes from thinking of the pair (mass, charge), hence the pair (rank, degree) as specifying a point in the plane

(M(E),Q(E))=(rank(E),degree(E)) 2 (M(E), \; Q(E)) \;=\; ( rank(E), \; degree(E) ) \;\in\; \mathbb{R}^2

Under this identification the charge density is the slope of a line of the line in the Cartesian plane which goes through zero and through (M(E),Q(E))(M(E), Q(E)), whence the term slope of a coherent sheaf. But understanding this not as a slope but as a charge density reveals why this has anything to do with “stability”, as we proceed to explain now.

First notice that, alternatively, we may identify the real plane with the complex plane and thus unify the mass and charge of D-branes into a single complex number

(1)Z(E) M(E)+iQ(E) =m(E)exp(iπϕ(E)) \begin{aligned} Z(E) & \coloneqq\; M(E) + i \, Q(E) \\ & = m(E) \, \exp( i \pi \phi(E) ) \;\in\; \mathbb{C} \end{aligned}

with an absolute value m(E)m(E) and a complex phase (modulo π\pi) ϕ(E)\phi(E).

(Such unification of two different quantities into a single complex quantity appears all over supersymmetric theory, for instance also in the definition of complex volume or the complex coupling constant appearing in the context of S-duality.)

Finally to see what all this has to do with “stability”:

In a supersymmetric theory such as superstring theory the stable states are supposed to be the BPS-states. When thinking of D-branes as black branes, being higher dimensional generalizations of charged black holes, the BPS states correspond to the higher dimensional analog of the extremal black holes, namely those that carry maximum charge for given mass, hence that maximize their charge density

stable D-braneBPS/extremal black D-brane maximum charge density Q/M \text{stable D-brane} \;\;\Leftrightarrow\;\; \text{BPS/extremal black D-brane} \;\;\Leftrightarrow\;\; \text{ maximum charge density }\, Q/M

Now it is plausible that a D-brane EE maximizes its potential charge density if removing any part ee of it reveals that ee by itself has lower charge density.

More formally, EE should maximize its charge density Q(E)/M(E)Q(E)/M(E) and hence be stable if for all sub-parts

eE e \subset E

hence for all subobjects in the relevant category (such as that of coherent sheaves) we have that the charge density of the part is smaller than the charge density of the whole:

Q(e)M(e)<Q(E)M(E) \frac{Q(e)}{M(e)} \;\lt\; \frac{ Q(E) }{ M(E) }

Conversely, this means that a D-brane state ee can increase its charge density, hence get close to being BPS and hence stable, by forming a bound state to be come an EE.

In the case of coherent sheaves, where the D-brane charge density is called the slope of a coherent sheaf slope(E)=Q(E)/M(E)slope(E) = Q(E)/M(E), as above, this says that EE is a stable coherent sheaf precisely if for all subobjects eEe \subset E we have

slope(e)<slope(E). slope(e) \;\lt\; slope(E) \,.

This is the classical formulation of slope stability or μ\mu-stability of coherent sheaves.

Of course we may equivalently re-express in terms of the complex phase ϕ(E)\phi(E) induced by the complexified mass/charge from (1). Since the slope of a complex number (regarded as a vector in the plane) equals the tan of its complex phase we have

Q(E)M(E)=tan(ϕ(E)/2). \frac{Q(E)}{M(E)} \;=\; tan\big( \phi(E)/2 \big) \,.

But since tan(()/2)tan((-)/2) is a monotone function on the domain (π/2,π/2)(-\pi/2, \pi/2), then if we agree to regard ϕ\phi as taking values in that interval, then the above stability condition becomes equivalently the statement that for all subobjects eEe \hookrightarrow E we have

ϕ(e)<ϕ(E). \phi(e) \;\lt\; \phi(E) \,.

This is the form in which Bridgeland 02 phrases the stability condition, see Def. below.

But the upshot is that this is still equivalent to saying that EE is stable precisely if it maximizes its charge- over mass-density, as befits an extremal black hole and, more generally, a BPS black brane.

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Definition

Definition

(stability function)

Let 𝒜\mathcal{A} be an additive category.

A stability function (sometimes also called a central charge) is a linear map

(2)Z:K(𝒜) Z \;\colon\; K(\mathcal{A}) \longrightarrow \mathbb{C}

from Grothendieck group K(𝒜)K(\mathcal{A}) to the additive group of complex numbers, such that for all non-zero objects E𝒜E \in \mathcal{A}, the image Z(E)Z(E) lies in the semi-upper half plane

H={rexp(iπϕ)|r>0and0<ϕ1} H \;=\; \big\{ r exp(i\pi \phi) \;\vert\; r \gt 0 \;\text{and}\; 0 \lt \phi \leq 1 \big\}

The phase

(3)ϕ(E)1πarg(Z(E))(0,1] \phi(E) \;\coloneqq\; \tfrac{1}{\pi} arg\big( Z(E) \big) \;\in\; (0,1]

of an non-zero object E𝒜E \in \mathcal{A} is just the complex phase 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.

(Bridgeland 02, Def. 2.1)

Definition

((semi-)stable objects)

For 𝒜\mathcal{A} an abelian category equipped with a stability function Z=rexp(iπϕ)Z = r exp(i\pi \phi) (Def. ). Then a non-zero object E𝒜E \in \mathcal{A} is called

  1. a semi-stable object if for all non-zero subobjects FEF\subset E the phase (3) of FF is smaller or equal that of EE

    (4)ϕ(F)ϕ(E) \phi(F) \;\leq\; \phi(E)
  2. a stable object if for all non-zero, proper subobjects FEF \subset E the phase (3) of FF is strictly smaller than that of EE:

    (5)ϕ(F)<ϕ(E). \phi(F) \;\lt\; \phi(E) \,.

(Bridgeland 02, Def. 2.2)

Definition

(Harder-Narasimhan property)

A stability function Z:K(𝒜)Z \;\colon\; K(\mathcal{A})\to \mathbb{C} (Def. ) 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=E 0=E_0 \subset E_1 \subset \cdots \subset E_n =E

such that the quotients

F i=E i/E i1 F_i = E_i/E_{i-1}

are all semi-stable (4) and satisfy

ϕ(F 1)>ϕ(F 2)>>ϕ(F n). \phi(F_1) \gt \phi(F_2) \gt \cdots \gt\phi(F_n) \,.

(Bridgeland 02, Def. 2.3)

Definition

(slicing)

Let 𝒟\mathcal{D} be a triangulated category (usually arising as the derived category of some abelian category).

A slicing 𝒫\mathcal{P} on 𝒟\mathcal{D} is a choice of additive full subcategories 𝒫(ϕ)𝒟\mathcal{P}(\phi) \subset \mathcal{D} 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>ϕ 2\phi_1 \gt \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>>ϕ n\phi_1 \gt \cdots \gt \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 mapping 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).

(Bridgeland 02, Def. 3.3)

Definition

(stability condition)

A stability condition on a triangulated category 𝒟\mathcal{D} is a pair σ=(Z,𝒫)\sigma = (Z, \mathcal{P}) consisting of

  1. a stability function (Def. )

  2. a slicing (Def. )

satisfying the relation that given a non-zero object E𝒫(ϕ)E\in \mathcal{P}(\phi), then there is a positive real number m(E)m(E) such that the value Z(E)Z(E) of the stability function (Def. ) is

Z(E)=m(E)exp(iπϕ). Z(E) \;=\; m(E) \, exp(i\pi \phi) \,.

(Bridgeland 02, Def. 5.1)

This justifies the repeated notation of ϕ\phi, since this says that if an object lies in a particular slice 𝒫(ϕ)\mathcal{P}(\phi), then it must also have complex phase ϕ\phi.

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Properties

In terms of hearts of t-structures

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

Schur’s lemma

Proposition

(Schur's lemma)

Let 𝒜\mathcal{A} be an abelian category equipped with a stability condition (Def. ).

For E𝒜E \in \mathcal{A} a stable object (Def ), every endomorphism of EE is either the zero morphism or is an isomorphism; in particular if 𝒜\mathcal{A} is enriched in vector spaces over an algebraically closed field kk, then End(E)kEnd(E) \simeq k.

More generally, for E 1,E 2𝒜E_1, E_2 \in \mathcal{A} two stable objects of the same slope/phase, ϕ(E 1)=ϕ(E 2)\phi(E_1) =\phi(E_2), any morphism E 1E 2E_1 \to E_2 is either the zero morphism or is an isomorphism.

e.g. (Bayer 11, 2.3 Exercise 1, Martinez 13, prop. 3)

Space of stability conditions

Proposition

(space of stability conditions)

Under reasonable hypotheses (…), one can put a natural topology on the space Stab(𝒟)Stab(\mathcal{D}) of all stability conditions (Def. ) , under which the space becomes a complex manifold.

(Bridgeland 02, theorem 1.2, reviewed in Bridgeland 09, 3.2)

Most work using this fact has been done in the case of the bounded derived category of coherent sheaves 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.

The space of stability conditions 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

Slope-stability of vector bundles / coherent sheaves

A motivating example for the concept of Bridgeland stability is the following classical notion.

Example

(slope-stability of coherent sheaves)

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.

In this case the standard stability function (Def. ) is

(6)Z(E)deg(E)+irk(E) Z(E) \;\coloneqq\; -deg(E) + i rk(E)

where degdeg is the degree and rkrk the rank of a coherent sheaf EE.

The classical notion of the slope of a vector bundle is

μ(E)rk(E)deg(E) \mu(E) \;\coloneqq\; \frac{rk(E)}{deg(E)}

When constructing a moduli space of vector bundles using GIT one needs to consider only slope (semi-)stable vector bundles (see e.g. Reineke 08, sections 3 and 4).

One can immediately see that a vector bundle/coherent sheaf is slope (semi-)stable if and only if it is (semi-)stable with respect to this stability function (6).

Thus Bridgeland stability generalizes the classical notions of stability of vector bundles.

Over resolutions of ADE-singularities

For G ADESU(2)G_{ADE} \subset SU(2) a finite subgroup of SU(2), let X˜\tilde X be the resolution of the corresponding ADE-singularity.

Then the connected component of the space of stability conditions (Def. ) on the bounded derived category of coherent sheaves over X˜\tilde X can be described explicitly (Thomas 02, Bridgeland 05, Brav-Thomas 09).

Specifically for type-A singularities the space of stability conditions (Prop. ) is in fact connected and simply-connected topological space (Ishii-Ueda-Uehara 10). In fact spaces of stability structures over Dynkin quivers are contractible (Qiu-Woold 14)

Brief review is in Bridgeland 09, section 6.3.

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References

General

The general definition is due to

generalizing the classical concept of slope-stability of vector bundles and of modules as in

  • Alastair King, Moduli of representations of finite dimensional algebras, The Quarterly Journal of Mathematics 45.4 (1994): 515-530 (pdf)

and motivated by informal arguments in string theory about the “Π\Pi-stability” for B-branes of the B-model topological string, due to Douglas-Fiol-Römerlsberger 00 and expanded on in Douglas 01, Douglas 02, Aspinwall 04 and other articles.

Further developments include

Introduction and review

  • Tom Bridgeland, Spaces of stability conditions, Proc. of symposia in pure math. 80, 2009 (math/0611510)

  • Markus Reineke, Moduli of representations of quivers (arXiv:0802.2147)

  • Arend Bayer, A tour to stability conditions on derived categories, 2011 (pdf)

  • Daniel Huybrechts, Introduction to stability conditions (arXiv:1111.1745)

  • Jan Engenhorst, Bridgeland Stability Conditions in Algebra, Geometry and Physics, 2014 (pdf)

  • Paolo Stellari, A tour on Bridgeland stability, 2015 (pdf)

Examples

Discussion of examples of stability conditions

over resolutions of ADE-singularities:

  • R. P. Thomas, Stability conditions and the braid group, Communications in Analysis and Geometry 14, 135-161, 2006 (arXiv:math/0212214)

  • Tom Bridgeland, Stability conditions and Kleinian singularities, International Mathematics Research Notices 2009.21 (2009): 4142-4157 (arXiv:0508257)

  • Akira Ishii, Kazushi Ueda, Hokuto Uehara, Stability conditions on A nA_n-singularities, Journal of Differential Geometry 84 (2010) 87-126 (arXiv:math/0609551)

  • Christopher Brav, Hugh Thomas, Braid groups and Kleinian singularities (arXiv:0910.2521)

  • Yu Qiu, Jon Woolf, Contractible stability spaces and faithful braid group actions (arXiv:1407.5986)

  • Yu Qiu, Def. 2.1 Stability conditions and quantum dilogarithm identities for Dynkin quivers, Adv. Math., 269 (2015), pp 220-264 (arXiv:1111.1010)

  • Tom Bridgeland, Yu Qiu, Tom Sutherland, Stability conditions and the A 2A_2 quiver (arXiv:1406.2566)

Relation to stable branes in string theory

The proposal that slope-stability of vector bundles should generalize to a notion of stability (“Π\Pi-stability”) of B-branes/D-branes originates with

In terms of stability (Π\Pi-stability) of B-branes of the B-model topological string:

On marginally stable branes:

Last revised on January 31, 2019 at 08:39:46. See the history of this page for a list of all contributions to it.