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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Bridgeland stability condition} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra} [[!include homological algebra - contents]] \hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable homotopy theory}}\label{stable_homotopy_theory} [[!include stable homotopy theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{AsStabilityOfBPSDBranes}{As stability of BPS D-branes}\dotfill \pageref*{AsStabilityOfBPSDBranes} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{in_terms_of_hearts_of_tstructures}{In terms of hearts of t-structures}\dotfill \pageref*{in_terms_of_hearts_of_tstructures} \linebreak \noindent\hyperlink{LemmaSchur}{Schur's lemma}\dotfill \pageref*{LemmaSchur} \linebreak \noindent\hyperlink{SpaceOfStabilityConditions}{Space of stability conditions}\dotfill \pageref*{SpaceOfStabilityConditions} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{SlopeStabilityOfVectorBundles}{Slope-stability of vector bundles / coherent sheaves}\dotfill \pageref*{SlopeStabilityOfVectorBundles} \linebreak \noindent\hyperlink{OverResolutionsOfADESingularities}{Over resolutions of ADE-singularities}\dotfill \pageref*{OverResolutionsOfADESingularities} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{general_2}{General}\dotfill \pageref*{general_2} \linebreak \noindent\hyperlink{introduction_and_review}{Introduction and review}\dotfill \pageref*{introduction_and_review} \linebreak \noindent\hyperlink{examples_2}{Examples}\dotfill \pageref*{examples_2} \linebreak \noindent\hyperlink{RelationToStableBranesInStringTheory}{Relation to stable branes in string theory}\dotfill \pageref*{RelationToStableBranesInStringTheory} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} \hypertarget{general}{}\subsubsection*{{General}}\label{general} A \emph{stability condition} (\hyperlink{Bridgeland02}{Bridgeland 02}) on a [[triangulated category]] (\href{triangulated+category#FromStableModelCategories}{hence} on a [[stable model category]]/[[stable (∞,1)-category]]) $\mathcal{D}$ (Def. \ref{StabilityCondition} below) singles out some of its [[objects]], to be called the ``stable objects'' (Def. \ref{StableObjects} below), that behave, to some extent, like [[indecomposable objects]] (whence ``stable''), in that a version of [[Schur's lemma]] applies to them (Prop. \ref{SchurLemma} below). Indeed, in the degenerate case that $\mathcal{D}$ is the [[derived category]] of a [[semisimple category]], the stable objects are just the [[irreducible objects]]. In the case where $\mathcal{D}$ is the bounded [[derived category]] of [[coherent sheaves]] on some [[complex manifold]], the general notion of stability reduces to the classical notion of [[slope of a coherent sheaf|slope]]-[[stable coherent sheaf|stability of coherent sheaves]] (``$\mu$-stability'', see Example \ref{SlopeStability} below). In other examples Bridgeland stability reproduces the derived analogue of [[David Mumford]]`s concept of [[GIT-stable point|stability]] in [[geometric invariant theory]] (\hyperlink{King94}{King 94}). There are in general different stability conditions on one and the same [[triangulated category]], in fact their [[moduli space]] $Stab(\mathcal{D})$ forms a [[complex manifold]] (Prop. \ref{SpaceOfStabilityConditions} below). The collection of stable objects in $\mathcal{D}$ 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 $Stab(\mathcal{D})$, called \emph{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 \emph{[[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 (\hyperlink{DouglasFiolRomerlsberger00}{Douglas-Fiol-Romerlsberger 00}) and followups (``$\Pi$-stability''). This [[string theory|string-theoretic]] interpretation also sheds light on the older notion of [[slope of a coherent sheaf|slope]]-[[stable coherent sheaf|stability of coherent sheaves]], we discuss this \hyperlink{AsStabilityOfBPSDBranes}{below}. \hypertarget{AsStabilityOfBPSDBranes}{}\subsubsection*{{As stability of BPS D-branes}}\label{AsStabilityOfBPSDBranes} 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 hole|extremal]] [[black branes]]. This yields a very simple and beautiful picture, which may be hard to extract from the original proposal due to \hyperlink{DouglasFiolRomerlsberger00}{Douglas-Fiol-Romerlsberger 00} (as witnessed by various reviews of the concept, see for instance \hyperlink{Stellari15}{Stellari 15, slide 9}). A key point to notice is that \emph{[[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 $X$ a [[space]], to be thought of as [[spacetime]], a [[coherent sheaf]] $E$ over $X$ may be thought of as an [[abelian sheaf]] of [[sections]] of a kind of [[complex vector bundle]] over $X$, whose [[fiber]] [[dimension]] is allowed to jump in some controlled way. Hence coherent sheaves are a slight generalization of \emph{[[complex vector bundles]]}. When considering [[D-branes]] on $X$, such coherent sheaves/vector bundles $E$ appear as the [[Chan-Paton gauge fields]] on the D-brane. \begin{displaymath} \text{object}\, E \phantom{AA}\leftrightarrow\phantom{AA} \text{D-brane carrying mass and charge} \end{displaymath} From this perspective, the [[rank of a vector bundle|rank]] \begin{displaymath} rank(E) \;\in\; \mathbb{R} \end{displaymath} of $E$ 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 $E$ is proportional to this rank. Ignoring the constant proportionality factor, we make this explicit by re-writing the rank as \begin{displaymath} M(E) \;\coloneqq\; rank(E) \,. \end{displaymath} In addition to their mass, the D-branes carry [[charge]], called [[RR-field|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]] $c_1(E)$. In generalization of this, the total [[D-brane charge]] reflected in a [[Chan-Paton gauge field]] $E$ is proportional its [[Chern character]] $ch(E)$. \begin{displaymath} Q(E) \;\in\; \mathbb{R} \,. \end{displaymath} For coherent sheaves this is essentially what is classically called the \emph{[[degree of a coherent sheaf]]}. Interpreting this as the [[charge]] carried by the D-brane, we may write \begin{displaymath} Q(E) \;=\; degree(E) \end{displaymath} Accordingly, [[D-branes]] $E$ have a \emph{charge density} proportional to \begin{displaymath} ChargeDensity(E) \;\coloneqq\; \frac{ Q(E) }{ M(E) } \;=\; \frac{ degree(E) }{ rank(E) } \;=\; slope(E) \,. \end{displaymath} This quotient is what is classically called the \emph{[[slope of a coherent sheaf|slope]]}, as shown on the right. The terminology comes from thinking of the [[pair]] ([[mass]], [[charge]]), hence the pair ([[rank of a coherent sheaf|rank]], [[degree of a coherent sheaf|degree]]) as specifying a [[point]] in the [[plane]] \begin{displaymath} (M(E), \; Q(E)) \;=\; ( rank(E), \; degree(E) ) \;\in\; \mathbb{R}^2 \end{displaymath} Under this identification the charge density is the [[slope of a line]] of the [[line]] in the [[Cartesian space|Cartesian]] [[plane]] which goes through [[zero]] and through $(M(E), Q(E))$, whence the term \emph{[[slope of a coherent sheaf]]}. But understanding this not as a slope but as a \emph{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 numbers|real]] [[plane]] with the [[complex plane]] and thus unify the [[mass]] and [[charge]] of [[D-branes]] into a single [[complex number]] \begin{equation} \begin{aligned} Z(E) & \coloneqq\; M(E) + i \, Q(E) \\ & = m(E) \, \exp( i \pi \phi(E) ) \;\in\; \mathbb{C} \end{aligned} \label{TheComplexifiedMassInIdeaSection}\end{equation} with an [[absolute value]] $m(E)$ and a [[complex phase]] (modulo $\pi$) $\phi(E)$. (Such unification of two different quantities into a single complex quantity appears all over [[supersymmetry|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 [[supersymmetry|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 \emph{[[extremal black holes]]}, namely those that carry maximum [[charge]] for given [[mass]], hence that \emph{maximize their charge density} \begin{displaymath} \text{stable D-brane} \;\;\Leftrightarrow\;\; \text{BPS/extremal black D-brane} \;\;\Leftrightarrow\;\; \text{ maximum charge density }\, Q/M \end{displaymath} Now it is plausible that a D-brane $E$ maximizes its potential charge density if removing any part $e$ of it reveals that $e$ by itself has lower charge density. More formally, $E$ should maximize its charge density $Q(E)/M(E)$ and hence be \emph{stable} if for all sub-parts \begin{displaymath} e \subset E \end{displaymath} hence for all \emph{[[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: \begin{displaymath} \frac{Q(e)}{M(e)} \;\lt\; \frac{ Q(E) }{ M(E) } \end{displaymath} Conversely, this means that a D-brane state $e$ can increase its charge density, hence get close to being BPS and hence stable, by forming a [[bound state]] to be come an $E$. In the case of coherent sheaves, where the D-brane charge density is called the \emph{[[slope of a coherent sheaf]]} $slope(E) = Q(E)/M(E)$, as above, this says that $E$ is a [[stable coherent sheaf]] precisely if for all [[subobjects]] $e \subset E$ we have \begin{displaymath} slope(e) \;\lt\; slope(E) \,. \end{displaymath} This is the classical formulation of \emph{[[stable coherent sheaf|slope stability]]} or \emph{$\mu$-stability} of [[coherent sheaves]]. Of course we may equivalently re-express in terms of the [[complex phase]] $\phi(E)$ induced by the complexified mass/charge from \eqref{TheComplexifiedMassInIdeaSection}. Since the slope of a [[complex number]] (regarded as a [[vector]] in the [[plane]]) equals the [[tan]] of its [[complex phase]] we have \begin{displaymath} \frac{Q(E)}{M(E)} \;=\; tan\big( \phi(E)/2 \big) \,. \end{displaymath} But since $tan((-)/2)$ is a [[monotone function]] on the [[domain]] $(-\pi/2, \pi/2)$, then if we agree to regard $\phi$ as taking values in that [[interval]], then the above stability condition becomes equivalently the statement that for all [[subobjects]] $e \hookrightarrow E$ we have \begin{displaymath} \phi(e) \;\lt\; \phi(E) \,. \end{displaymath} This is the form in which \hyperlink{Bridgeland02}{Bridgeland 02} phrases the stability condition, see Def. \ref{StableObjects} below. But the upshot is that this is still equivalent to saying that $E$ is stable precisely if it maximizes its [[charge]]- over [[mass]]-density, as befits an [[extremal black hole]] and, more generally, a [[BPS state|BPS]] [[black brane]]. $\,$ \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \begin{defn} \label{StabilityFunction}\hypertarget{StabilityFunction}{} \textbf{(stability function)} Let $\mathcal{A}$ be an [[additive category]]. A \textbf{stability function} (sometimes also called a \emph{central charge}) is a [[linear map]] \begin{equation} Z \;\colon\; K(\mathcal{A}) \longrightarrow \mathbb{C} \label{StabilityFunction}\end{equation} from [[Grothendieck group]] $K(\mathcal{A})$ to the [[abelian group|additive group]] of [[complex numbers]], such that for all non-[[zero objects]] $E \in \mathcal{A}$, the [[image]] $Z(E)$ lies in the [[upper half-plane|semi-upper half plane]] \begin{displaymath} H \;=\; \big\{ r exp(i\pi \phi) \;\vert\; r \gt 0 \;\text{and}\; 0 \lt \phi \leq 1 \big\} \end{displaymath} The \emph{phase} \begin{equation} \phi(E) \;\coloneqq\; \tfrac{1}{\pi} arg\big( Z(E) \big) \;\in\; (0,1] \label{PhaseOfAnObject}\end{equation} of an non-[[zero object]] $E \in \mathcal{A}$ is just the [[complex phase]] $\phi$ that occurs in the representation from $H$. Alternatively, by plotting $Z(E)$ in the [[complex plane]], the phase is the argument (slope) divided by $\pi$. \end{defn} (\hyperlink{Bridgeland02}{Bridgeland 02, Def. 2.1}) \begin{defn} \label{StableObjects}\hypertarget{StableObjects}{} \textbf{((semi-)stable objects)} For $\mathcal{A}$ an [[abelian category]] equipped with a stability function $Z = r exp(i\pi \phi)$ (Def. \ref{StabilityFunction}). Then a non-[[zero object]] $E \in \mathcal{A}$ is called \begin{enumerate}% \item a \emph{semi-stable object} if for all non-[[zero object|zero]] [[subobjects]] $F\subset E$ the phase \eqref{PhaseOfAnObject} of $F$ is smaller or equal that of $E$ \begin{equation} \phi(F) \;\leq\; \phi(E) \label{SemiStability}\end{equation} \item a \emph{stable object} if for all non-[[zero object|zero]], proper [[subobjects]] $F \subset E$ the phase \eqref{PhaseOfAnObject} of $F$ is strictly smaller than that of $E$: \begin{equation} \phi(F) \;\lt\; \phi(E) \,. \label{Stability}\end{equation} \end{enumerate} \end{defn} (\hyperlink{Bridgeland02}{Bridgeland 02, Def. 2.2}) \begin{defn} \label{HarderNarasimhanProperty}\hypertarget{HarderNarasimhanProperty}{} \textbf{(Harder-Narasimhan property)} A stability function $Z \;\colon\; K(\mathcal{A})\to \mathbb{C}$ (Def. \ref{StabilityFunction}) is said to have the \emph{Harder-Narasimhan property} if for any non-[[zero object]] $E$ there exists a finite [[filtered object|filtration]] by [[subobjects]] \begin{displaymath} 0=E_0 \subset E_1 \subset \cdots \subset E_n =E \end{displaymath} such that the [[quotients]] \begin{displaymath} F_i = E_i/E_{i-1} \end{displaymath} are all semi-stable \eqref{SemiStability} and satisfy \begin{displaymath} \phi(F_1) \gt \phi(F_2) \gt \cdots \gt\phi(F_n) \,. \end{displaymath} \end{defn} (\hyperlink{Bridgeland02}{Bridgeland 02, Def. 2.3}) \begin{defn} \label{Slicing}\hypertarget{Slicing}{} \textbf{(slicing)} Let $\mathcal{D}$ be a [[triangulated category]] (usually arising as the [[derived category]] of some [[abelian category]]). A \emph{slicing} $\mathcal{P}$ on $\mathcal{D}$ is a choice of [[additive category|additive]] [[full subcategories]] $\mathcal{P}(\phi) \subset \mathcal{D}$ for each $\phi \in \mathbb{R}$ satisfying \begin{enumerate}% \item $\mathcal{P}(\phi +1)=\mathcal{P}(\phi)[1]$ \item If $\phi_1 \gt \phi_2$ and $A_j\in \mathcal{P}(\phi_j)$, then $Hom(A_1, A_2)=0$. \item Any object has a finite [[filtration]] by the slicing: If $E\in \mathcal{D}$, then there exists $\phi_1 \gt \cdots \gt \phi_n$ and a sequence $0=E_0\to E_1 \to \cdots \to E_n = E$ such that the [[mapping cone]] $E_{j-1}\to E_j \to F_j \to E_{j-1}[1]$ satisfies $F_j\in \mathcal{P}(\phi_j)$. \end{enumerate} \end{defn} (\hyperlink{Bridgeland02}{Bridgeland 02, Def. 3.3}) \begin{defn} \label{StabilityCondition}\hypertarget{StabilityCondition}{} \textbf{(stability condition)} A \textbf{stability condition} on a [[triangulated category]] $\mathcal{D}$ is a [[pair]] $\sigma = (Z, \mathcal{P})$ consisting of \begin{enumerate}% \item a stability function (Def. \ref{StabilityFunction}) \item a slicing (Def. \ref{Slicing}) \end{enumerate} satisfying the relation that given a non-[[zero object]] $E\in \mathcal{P}(\phi)$, then there is a [[positive number|positive]] [[real number]] $m(E)$ such that the value $Z(E)$ of the stability function (Def. \ref{StabilityCondition}) is \begin{displaymath} Z(E) \;=\; m(E) \, exp(i\pi \phi) \,. \end{displaymath} \end{defn} (\hyperlink{Bridgeland02}{Bridgeland 02, Def. 5.1}) This justifies the repeated notation of $\phi$, since this says that if an object lies in a particular slice $\mathcal{P}(\phi)$, then it must also have [[complex phase]] $\phi$. $\,$ \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{in_terms_of_hearts_of_tstructures}{}\subsubsection*{{In terms of hearts of t-structures}}\label{in_terms_of_hearts_of_tstructures} Bridgeland proves that an equivalent way to give a stability is to specify a heart of a bounded [[t-structure]] on $\mathcal{D}$ and give a stability function the heart that satisfies the Harder-Narasimhan property. \hypertarget{LemmaSchur}{}\subsubsection*{{Schur's lemma}}\label{LemmaSchur} \begin{prop} \label{SchurLemma}\hypertarget{SchurLemma}{} \textbf{([[Schur's lemma]])} Let $\mathcal{A}$ be an [[abelian category]] equipped with a stability condition (Def. \ref{StabilityCondition}). For $E \in \mathcal{A}$ a stable object (Def \ref{StableObjects}), every [[endomorphism]] of $E$ is either the [[zero morphism]] or is an [[isomorphism]]; in particular if $\mathcal{A}$ is [[enriched category|enriched]] in [[vector spaces]] over an [[algebraically closed field]] $k$, then $End(E) \simeq k$. More generally, for $E_1, E_2 \in \mathcal{A}$ two stable objects of the same slope/phase, $\phi(E_1) =\phi(E_2)$, any [[morphism]] $E_1 \to E_2$ is either the [[zero morphism]] or is an [[isomorphism]]. \end{prop} e.g. (\hyperlink{Bayer11}{Bayer 11, 2.3 Exercise 1}, \hyperlink{Martinez13}{Martinez 13, prop. 3}) \hypertarget{SpaceOfStabilityConditions}{}\subsubsection*{{Space of stability conditions}}\label{SpaceOfStabilityConditions} \begin{prop} \label{SpaceOfStabilityConditions}\hypertarget{SpaceOfStabilityConditions}{} \textbf{(space of stability conditions)} Under reasonable hypotheses (\ldots{}), one can put a natural [[topological manifold|topology]] on the space $Stab(\mathcal{D})$ of all stability conditions (Def. \ref{StabilityCondition}) , under which the space becomes a [[complex manifold]]. \end{prop} (\hyperlink{Bridgeland02}{Bridgeland 02, theorem 1.2}, reviewed in \hyperlink{Bridgeland09}{Bridgeland 09, 3.2}) Most work using this fact has been done in the case of the [[bounded derived category of coherent sheaves]] $\mathcal{D}=D^b(Coh(X))$ where $X$ is a smooth, projective [[algebraic variety|variety]] over $\mathbb{C}$ so that $\mathcal{D}$ is $\mathbb{C}$-linear and $K(\mathcal{D})$ is finitely generated. The space of stability conditions $Stab(X)$ has a locally finite wall and chamber decomposition. The philosophy is that if one fixes a numerical condition $v$ and considers the [[moduli space]] of $\sigma$-stable sheaves as $\sigma$ varies through $Stab(X)$, then the moduli spaces $M_\sigma(v)\simeq M_{\sigma '}(v)$ should be isomorphic if $\sigma$ and $\sigma'$ are in the same chamber. Bayer and Macri have shown this is true for K3 surfaces, and upon [[wall crossing|crossing a wall]] the moduli spaces are related by a [[birational map]]. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{SlopeStabilityOfVectorBundles}{}\subsubsection*{{Slope-stability of vector bundles / coherent sheaves}}\label{SlopeStabilityOfVectorBundles} A motivating example for the concept of Bridgeland stability is the following classical notion. \begin{example} \label{SlopeStability}\hypertarget{SlopeStability}{} \textbf{([[slope of a coherent sheaf|slope]]-[[stable coherent sheaf|stability of coherent sheaves]])} Let $X$ be a non-singular, [[algebraic curve|projective curve]] over $\mathbb{C}$. Let $\mathcal{A}=Coh(X)$ be the [[category]] of [[coherent sheaves]] on $X$. In this case the standard stability function (Def. \ref{StabilityFunction}) is \begin{equation} Z(E) \;\coloneqq\; -deg(E) + i rk(E) \label{StandardStabilityFunctionForVectorBundles}\end{equation} where $deg$ is the \emph{[[degree of a coherent sheaf|degree]]} and $rk$ the \emph{[[rank of a coherent sheaf|rank]]} of a [[coherent sheaf]] $E$. The classical notion of the \emph{[[slope of a coherent sheaf|slope]] of a vector bundle is} \begin{displaymath} \mu(E) \;\coloneqq\; \frac{rk(E)}{deg(E)} \end{displaymath} When constructing a [[moduli space]] of vector bundles using [[geometric invariant theory|GIT]] one needs to consider only [[slope of a coherent sheaf|slope]] (semi-)[[stable vector bundles]] (see e.g. \hyperlink{Reineke08}{Reineke 08, sections 3 and 4}). One can immediately see that a [[vector bundle]]/[[coherent sheaf]] is [[slope of a vector bundle|slope]] (semi-)[[stable vector bundle|stable]] if and only if it is (semi-)stable with respect to this stability function \eqref{StandardStabilityFunctionForVectorBundles}. \end{example} Thus Bridgeland stability generalizes the classical notions of [[stable vector bundle|stability of vector bundles]]. \hypertarget{OverResolutionsOfADESingularities}{}\subsubsection*{{Over resolutions of ADE-singularities}}\label{OverResolutionsOfADESingularities} For $G_{ADE} \subset SU(2)$ a [[finite subgroup of SU(2)]], let $\tilde X$ be the [[resolution of singularities|resolution]] of the corresponding [[ADE-singularity]]. Then the [[connected component]] of the space of stability conditions (Def. \ref{StabilityCondition}) on the bounded [[derived category]] of [[coherent sheaves]] over $\tilde X$ can be described explicitly (\hyperlink{Thomas02}{Thomas 02}, \hyperlink{Bridgeland05}{Bridgeland 05}, \hyperlink{BravThomas09}{Brav-Thomas 09}). Specifically for type-A singularities the space of stability conditions (Prop. \ref{SpaceOfStabilityConditions}) is in fact [[connected topological space|connected]] and [[simply-connected topological space]] (\hyperlink{IshiiUedaUehara10}{Ishii-Ueda-Uehara 10}). In fact spaces of stability structures over [[Dynkin quivers]] are [[contractible space|contractible]] (\hyperlink{QiuWoold14}{Qiu-Woold 14}) Brief review is in \hyperlink{Bridgeland09}{Bridgeland 09, section 6.3}. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[BPS states]] \item [[wall crossing]] \item [[geometric invariant theory]] \end{itemize} $\,$ \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{general_2}{}\subsubsection*{{General}}\label{general_2} The general definition is due to \begin{itemize}% \item [[Tom Bridgeland]], \emph{Stability conditions on triangulated categories}, Ann. of Math. 166 (2007) 317--345 (\href{http://arxiv.org/abs/math/0212237}{math.AG/0212237}) \end{itemize} generalizing the classical concept of [[slope of a vector bundle|slope]]-[[stable vector bundle|stability of vector bundles]] and of [[modules]] as in \begin{itemize}% \item [[Alastair King]], \emph{Moduli of representations of finite dimensional algebras}, The Quarterly Journal of Mathematics 45.4 (1994): 515-530 (\href{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.623.649&rep=rep1&type=pdf}{pdf}) \end{itemize} and motivated by informal arguments in [[string theory]] about the ``$\Pi$-stability'' for [[B-branes]] of the [[B-model]] [[topological string]], due to \hyperlink{DouglasFiolRomerlsberger00}{Douglas-Fiol-Römerlsberger 00} and expanded on in \hyperlink{Douglas01}{Douglas 01}, \hyperlink{Douglas02}{Douglas 02}, \hyperlink{Aspinwall04}{Aspinwall 04} and other articles. Further developments include \begin{itemize}% \item R. Pandharipande, R.P. Thomas, \emph{Stable pairs and BPS invariants}, \href{http://arxiv.org/abs/0711.3899}{arXiv:0711.3899} \item [[Maxim Kontsevich]], [[Yan Soibelman]], \emph{Stability structures, motivic Donaldson-Thomas invariants and cluster transformations}, \href{http://arxiv.org/abs/0811.2435}{arXiv:0811.2435} \item Rina Anno, Roman Bezrukavnikov, Ivan Mirkovi, \emph{A thin stringy moduli space for Slodowy slices}, \href{http://arxiv.org/abs/1108.1563}{arxiv/1108.1563} \item Arend Bayer, Emaneule Macri, \emph{Projectivity and Birational Geometry of Bridgeland Moduli Spaces} (\href{http://arxiv.org/abs/1203.4613}{arXiv:1203.4613}) \item [[Tom Bridgeland]], Ivan Smith, \emph{Quadratic differentials as stability conditions}, \href{http://arxiv.org/abs/1302.7030}{arxiv/1302.7030} \item Cristian Martinez, \emph{Duality, Bridgeland wall-crossing and flips of secant varieties} (\href{https://arxiv.org/abs/1311.1183}{arXiv:1311.1183}) \end{itemize} \hypertarget{introduction_and_review}{}\subsubsection*{{Introduction and review}}\label{introduction_and_review} \begin{itemize}% \item [[Tom Bridgeland]], \emph{Spaces of stability conditions}, Proc. of symposia in pure math. \textbf{80}, 2009 (\href{http://arxiv.org/abs/math/0611510}{math/0611510}) \item Markus Reineke, \emph{Moduli of representations of quivers} (\href{https://arxiv.org/abs/0802.2147}{arXiv:0802.2147}) \item Arend Bayer, \emph{A tour to stability conditions on derived categories}, 2011 (\href{https://www.maths.ed.ac.uk/~abayer/dc-lecture-notes.pdf}{pdf}) \item [[Daniel Huybrechts]], \emph{Introduction to stability conditions} (\href{https://arxiv.org/abs/1111.1745}{arXiv:1111.1745}) \item Jan Engenhorst, \emph{Bridgeland Stability Conditions in Algebra, Geometry and Physics}, 2014 (\href{https://www.freidok.uni-freiburg.de/fedora/objects/freidok:9595/datastreams/FILE1/content}{pdf}) \item Paolo Stellari, \emph{A tour on Bridgeland stability}, 2015 (\href{https://users.unimi.it/stellari/Research/Slides/LucidiStabCondBeamer.pdf}{pdf}) \end{itemize} \hypertarget{examples_2}{}\subsubsection*{{Examples}}\label{examples_2} Discussion of examples of stability conditions over [[resolution of singularities|resolutions]] of [[ADE-singularities]]: \begin{itemize}% \item R. P. Thomas, \emph{Stability conditions and the braid group}, Communications in Analysis and Geometry 14, 135-161, 2006 (\href{https://arxiv.org/abs/math/0212214}{arXiv:math/0212214}) \item [[Tom Bridgeland]], \emph{Stability conditions and Kleinian singularities}, International Mathematics Research Notices 2009.21 (2009): 4142-4157 (\href{https://arxiv.org/abs/math/0508257}{arXiv:0508257}) \item Akira Ishii, Kazushi Ueda, Hokuto Uehara, \emph{Stability conditions on $A_n$-singularities}, Journal of Differential Geometry 84 (2010) 87-126 (\href{https://arxiv.org/abs/math/0609551}{arXiv:math/0609551}) \item [[Christopher Brav]], Hugh Thomas, \emph{Braid groups and Kleinian singularities} (\href{https://arxiv.org/abs/0910.2521}{arXiv:0910.2521}) \item [[Yu Qiu]], Jon Woolf, \emph{Contractible stability spaces and faithful braid group actions} (\href{https://arxiv.org/abs/1407.5986}{arXiv:1407.5986}) \item [[Yu Qiu]], Def. 2.1 \emph{Stability conditions and quantum dilogarithm identities for Dynkin quivers}, Adv. Math., 269 (2015), pp 220-264 (\href{https://arxiv.org/abs/1111.1010}{arXiv:1111.1010}) \item [[Tom Bridgeland]], [[Yu Qiu]], Tom Sutherland, \emph{Stability conditions and the $A_2$ quiver} (\href{https://arxiv.org/abs/1406.2566}{arXiv:1406.2566}) \end{itemize} \hypertarget{RelationToStableBranesInStringTheory}{}\subsubsection*{{Relation to stable branes in string theory}}\label{RelationToStableBranesInStringTheory} The proposal that slope-stability of vector bundles should generalize to a notion of stability (``$\Pi$-stability'') of [[B-branes]]/[[D-branes]] originates with \begin{itemize}% \item [[Michael Douglas]], Bartomeu Fiol, Christian Römelsberger, \emph{Stability and BPS branes}, JHEP 0509:006, 2005 (\href{https://arxiv.org/abs/hep-th/0002037}{arXiv:hep-th/0002037}) \end{itemize} In terms of stability ($\Pi$-stability) of [[B-branes]] of the [[B-model]] [[topological string]]: \begin{itemize}% \item [[Michael Douglas]], \emph{D-branes, categories and $N=1$ supersymmetry, J.Math.Phys. \textbf{42} (2001) 2818--2843;} \item [[Michael Douglas]], \emph{Dirichlet branes, homological mirror symmetry, and stability}, Proc. ICM, Vol. III (Beijing, 2002), 395--408, Higher Ed. Press, Beijing, 2002 \item [[Paul Aspinwall]], \emph{D-Branes on Calabi-Yau Manifolds} (\href{https://arxiv.org/abs/hep-th/0403166}{arXiv:hep-th/0403166}) \item [[Aaron Bergman]], \emph{Stability Conditions and Branes at Singularities}, Journal of High Energy Physics 2008.10 (2008): 07 (\href{http://arxiv.org/abs/hep-th/0702092}{arXiv:hep-th/0702092}) \item Dmitry Malyshev, [[Herman Verlinde]], \emph{D-branes at Singularities and String Phenomenology}, Nucl.Phys.Proc.Suppl.171:139-163, 2007 (\href{https://arxiv.org/abs/0711.2451}{arXiv:0711.2451}) \end{itemize} On marginally stable branes: \begin{itemize}% \item [[Paul Aspinwall]], Alexander Maloney, Aaron Simons, \emph{Black Hole Entropy, Marginal Stability and Mirror Symmetry}, JHEP0707:034, 2007 (\href{https://arxiv.org/abs/hep-th/0610033}{arxiv:hep-th/0610033}) \end{itemize} [[!redirects Bridgeland stability conditions]] [[!redirects Bridgeland stability]] [[!redirects stability condition]] [[!redirects stability conditions]] [[!redirects stability function]] [[!redirects stability functions]] \end{document}