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There was an “Idea” at Bridgeland stability condition. I added the sections Definition, Key Results, and Examples. I also added a reference for the last key result listed. I should probably fill in some related stubs like t-structure or classical notions of stability.
I have added to the Idea-section at Bridgeland stability condition the original motivation as a formalization of reaction processes of D-branes for the topological B-model string. (A similar paragraph appears now also as an example in the entry chemical reaction.)
What’s the imagined scope of this theory? I remember you introduced it on the chemistry entry. Is there now a sense of the range of reaction/scattering processes that it might concern?
I see mathematicians are using the language of ’scattering diagrams’ in what seems a pure maths setting, as in Scattering diagrams, Hall algebras and stability conditions:
to give a categorical description of the scattering diagram one does not need to use triangulated categories. In fact, for the most part, we work in the abelian category of representations of a fixed quiver with relations…
You quickly reach a range of terms I’ve heard for a while without knowing what they’re really about: cluster, tilting, wall-crossing.
After thinking about it, I felt yesterday that the idea that is offered in the literature for what stability conditions really do is suboptimal, and that the following is the simple picture to have in mind:
We are talking about extended objects $E$ that have a mass $M(E)$ proportional to their volume (tension times their volume) and which carry some charge $Q(E)$ generalizing magnetic charge. Accordingly, they have a charge density proportional to
$ChargeDensity(E) \;\coloneqq\; Q(E)/M(E) ,.$For instance if $E$ is a vector bundle, thought of as the gauge bundle on a D-brane locus, then its rank is the number of coincident D-branes at that loci, each of which has the same tension, so that, up to some proportionality factor
$M(E) \;=\; rank(E)$in this case. Moreover, these D-branes carry a charge measured by their Chern-character. In lowest degree this is their first Chern class $c_1(E)$, which is literally the magnetic charge they carry, as identified by Dirac in 1930:
$Q(E) \;=\; \int c_1(E) \,.$So we learn that what is known as the slope of a vector bundle is really the charge density that it represents when regarded as a D-brane Chan-Paton gauge field
$ChargeDensity(E) \;\coloneqq\; Q(E)/M(E) \;=\; Slope(E) \,.$This is really what Bridgeland stability functions generalize, assignment of charge and mass:
$Z(E) \;=\; ( Q(E), M(E) ) \in \mathbb{R}^2 \,.$Now, to see what this has to do with stability:
The stable branes (BPS states) are supposed to be those that have maximal charge for given mass, hence the stable objects are those that maximize charge density. (The “extremal black holes”) So if
$e \subset E$is a subobject with
$\frac{Q(e)}{M(e)} \lt \frac{ Q(E) }{ M(E) }$this means that $e$ can increase its charge density by forming a bound state to be come an $E$.
So $e$ is less stable when all by itself than as a constituent of $E$, hence that $E$ cannot be further stabilized by splitting off an $e$.
That’s the condition of $\mu$-stability/$\Pi$-stability/Bridgeland stability.
And does it happen that that $E$ is not contained as a subobject in some object of higher charge density? What happens dynamically then, if this isn’t too naive a question? Is it that the number of coincident D-branes remains constant, or fluctuates about this maximum?
Something particularly nice happens in Dynkin quiver-like situations because of Gabriel’s theorem?
In that story, $E$ could itself eventually fuse into some bigger $\mathcal{E}$, if it runs into the relevant objects to form a bound state with. But, read the other way around, $E$ by itself won’t spontaneously decompose into its constituents as long as these, by themselves, have lower charge density than they have when all bound together as an $E$.
Gabriel’s theorem is about there being a finite number of elementary constitutents.
But coming back to my #6:
I thought you might show some reaction to being handed such a clear motivation of stability conditions, in comparison to the oft-cited but ever mysterious DFR 00 and its followups (had to laugh when I saw people cite Wittgenstein in reaction, slide 9 here)
Certainly your account has an admirably clarity to it. But I think I don’t know enough of the terrain to have any proper sense of what there is to be clarified.
I take it this line of research emerged out of mirror symmetry as described here by Verma:
Kontsevich’s Mirror symmetry states that two Calabi-Yau manifolds are mirror to each other if the derived category of coherent sheaves of one is equivalent to the derived Fukaya category of the other. In physics, this means that the D-branes for Type IIA string theory compactified on a CY manifold is equivalent to the D-branes for Type IIB string theory compactified on the mirror CY manifold. (D-branes are ’supposed’ be the objects of the derived categories). Of special interest to physicists are the so-called BPS D-branes, which Douglas argued are the $\pi$-stable D-branes.
Now the question is - What are the mathematical counterparts of BPS D-branes? Bridgeland’s stability is an attempt to define mathematically the notion of $\pi$-stability for the corresponding derived categories.
But what is the significance of BPS branes or the corresponding objects in the derived categories ? Mirror symmetry transcends to the statement that the subcategories of BPS branes at each point in the moduli space are equivalent for mirror manifolds. i.e. the subcategories of Bridgeland stable objects in the derived category of coherent sheaves on a CY manifold is equivalent to the ’stable’ objects in the derived Fukaya category. This plays important role in our understanding of the moduli space of the Calabi-Yau manifold (corresponding string theory). The BPS branes (and the stable objects in the derived category) also show a beautiful behavior called Wall-Crossing.
So you’re giving us a way to consider those BPS states.
Slide 38 of the talk you just linked to, says
Bridgeland stability is preserved under Fourier–Mukai equivalences
so that’s behind Verma’s comment.
And the stability idea has taken on a life of its own in pure mathematics, especially concerning relating compactifications of moduli spaces (as in the comment above Verma’s).
Well, I know more now than I did this morning!
I’d like to come back to my quest of figuring out the simple case of stability conditions and (marginally) stable objects on semisimple categories, such as $G Rep$.
I am suspecting that if $\mathcal{A}$ is a semisimple category, then for every choice of stability function
$(Q,M) \;\colon\; K(\mathcal{A}) \longrightarrow \mathbb{R}^2$the only strictly stable objects are the simple objects;
the marginally stable objects are the direct sums of simple objects all of whose direct summands have the same charge density charge and mass.
This boils down to checking the following elementary arithmetic:
Is it true that…
whenever
$\frac{ Q_i }{ M_i } \;\leq\; \frac{ Q_1 + Q_2 }{ M_1 + M_2 }$
for $i =1$ and $i = 2$, subject to $\frac{1}{\pi}arg( Q_i, M_i ) \in (0,1]$,
then in fact $Q_1 = Q_2$ and $M_1 = M_2$ and equality holds.
??
There was a little logical glitch in the sequence of definitions, dating back all the way to rev 4:
In the entry, a stability function was first defined on an abelian category, but then eventually used in the context of triangulated categories.
I made the first “abelian category” read “additive category” now, to fix this. But this may need attention not to conflict with some conventions/assumptions later on.
Typos
such that for all non-zero objects $E \subset \mathcal{A}$,
and
Then the connected component of the space spring of stability conditions
I believe that ’spring’ is an aide-mémoire for Urs to return, as explained here.
Yes. Thanks, fixed now.
okay, I have now written a fair bit of an informal explanation of Bridgeland stability as Stability of D-branes.
It seems there is considerable demand for such an explanation, not just voiced in various reviews of the subject. such as Stellari 15, slide 9, but also seen on various discussion forums:
That is certainly clear!
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
Even if pure mathematics tends to obscure the physical motivation, does it suggest that there might be nearby parallels with other cohomologies playing the role of K-theory and other geometric figures than D-branes? I’m guessing that you’re hoping for some M-theory-cohomotopy type variant.
I wonder if one tracked back these ideas what the origins are. It seems Bridgeland stability owes something to Mumford’s stability in geometric invariant theory, which in turn was a revival of Hilbert’s work from the 1890s. Looking at the wikipedia page, that condition on degree over rank has been around a while.
But then out of Hilbert’s invariant theory eventually comes Noether and her theorems.
I got interested in this here because for half a week I mistakenly thought that equivariant stable cohomotopy is not surjecting onto the relevant equivariant K-theory group. I was trying to see if maybe it surjects onto the BPS-objects inside the K-theory group, which would make good physical sense, since the stability of the non-BPS D-branes showing up in the K-theory classifcation has only been checked in perturbative string theory.
But then I came to realize that the relevant fractional D-branes are really all Bridgeland-stable, on very general grounds, but also that the comparison map from equivariant stable cohomotopy is not supposed ot surject on equivariant KU, but on equivariant KO – which it does.
So this means that now, for the moment, my little excursion into stability conditions is over and I will be focusing again on other things now. But as an afterthought, before leaving the subject for the time being, I did want to record what no source before seems to really have offered, the simple conceptual idea behind the formalization of stability conditions.
Regarding Tom Bridgeland’s inspiration: I have no deeper insight into the history of this, but the Bridgeland stability condition on an additive category is almost verbatim that of the classical concept of slope stability (=$\mu$-stability) of coherent sheaves, which old articles like King’s explain how to relate to GIT stability. Therefore I suppose the key inspiration that Bridgeland took from Douglas’s writings is not so much the formulation of stability as such, but the idea that it should be applied to derived categories, hence, as we would say around here, to stable $\infty$-categories. So that’s what he did
Since stability depends only on comparing to subobjects, you could have a stable state $E$ which is a subobject of another stable state of higher slope $F$? The stability is against decomposition rather than augmentation?
We touched on this before. The Bridgeland-Schur lemma here says this can not happen if the larger stable object has the same slope as the smaller stable object. But of course one would be interested in this for different slopes anyway. Then I don’t know if there is a general statement like that. I’d need to look at more examples to get a feeling for this.
the comparison map from equivariant stable cohomotopy is not supposed ot surject on equivariant KU, but on equivariant KO – which it does.
have you written about this yet?
I am in the process of writing it up. Will give a talk on it next week, should have some notes to share by then.
Is there a reason why in equation (1) it’s Z = rank + i degree, whereas in (6) it’s Z = - degree + i rank? A $\pi/2$ phase shift.
Yes, I was going to add a paragraph on this but then didn’t get around to.
In what I wrote in the Idea-section, towards the end, we find that $\pi \phi$ has to be in $(-\pi/2, \pi/2)$, because that’s a maximal interval of definition of the tangent $tan(\phi) = \frac{sin(\phi)}{cos(\phi)} = \frac{1}{m(E)} \frac{Q(E)}{M(E)}$. But by just rotating the coordinate chart by 90 degrees one can of course pass to shifted conventions, where $\pi \phi$ is in $(0, \pi)$. This is what you see in Tom Bridgeland’s article, and many/all authors following him.
To come back to #24, #25, I can tell you about the core of the statement right away:
I am looking at the canonical comparison morphism
$\array{ \mathbb{S}^{\mathbb{H}}_{G_{DE} \times G^'_{DE}}(\mathbb{R}^{10,1}) \simeq \mathbb{S}^{\mathbb{H}}_{G_{DE} \times G^'_{DE}}(\ast) &\overset{\phantom{AA}\alpha\phantom{AA}}{\longrightarrow}& \mathbb{S}^{\left(\mathbb{H}^{G^'_{DE}}\right)}_{G_{DE}}(\ast) = \mathbb{S}^{0}_{G_{DE}}(\ast) &\overset{\phantom{AA}\beta\phantom{AA}}{\longrightarrow}& \mathbb{KO}^{0}_{G_{DE}}(\ast) \simeq R_{\mathbb{R}}(G_{DE}) \\ { \text{charge group} \atop \text{ of intersecting MK6-branes } } && { \text{charge lattice} \atop {\text{of fractional M-branes} \atop \text{ at MK6-singularity } } } && { \text{charge lattice} \atop {\text{of fractional D-branes} \atop \text{ at D-E-type singularity }} } }$from D-E-type equivariant cohomotopy in RO-degree $\mathbb{H}$ to equivariant KO-theory, of the point. This is supposed to be the map from the M-theory charges at the intersection point of two MK6-branes (11d KK-monopoles), where the first map $\alpha$ zooms in on one of the two MK6-singularities and regards the remaining one intersecting there, while the second map $\beta$ is the forgetful map “from M-brane charges to D-brane charges” at the singularity.
The cokernel of the total composite would be the fractional D-branes that do not lift to M-theory, while the kernel would be the extra M-theory degrees of freedom that are invisible in perturbation theory.
So to check that this is consistent with folklore, I need that whatever I find in the cokernel has a good reason to be non-perturbatively unstable. Whence the stability conditions in this thread here. But the conclusion is that everything in equivariant K-theory of a point should be stable (i.e. the irreps stable, their direct sums semi-stable). This means that the composite map ought to be surjective as long as we want to find compatibility with folklore (once we have found enough of that, we may turn this around, but maybe not yet).
But for various reasons the situation of interest is really the orientifold as opposed to plain orbifold case, which here means D-type and E-type singularities, but not A-type. For these folklore has it that we land in equivariant KO, hence in the real representation ring, as shown above.
Now
1) the first map, $\alpha$, is always surjective, I think, see the proof here;
1) to see that the second map $\beta$ is surjective onto the real representation for D-E-type finite subgroups, we use the help of some computer algebra provided by Simon Burton (who just have left Sydney for London). That computes the Burnside ring multiplication table, then puts it in Hermite normal form whose rows, one can prove, form a basis of the image of $\beta$. So to see that $\beta$ is surjective one can now compare to character tables in the literature.
(beta is not surjective on the complex irreps, due to non-trivial Schur decomposition, hence the analog of the above map to $KU^0_{G_{DE}}(\ast)$ is not surjective. For about one week I tried to see if the direct summands of complex irreps that a rational/real rep decomposes into are maybe not “stable” by themselves, but only as compounds of the bigger rational/real irrep. But that’s now abandoned. )
I was just trying to get some small sense of how the ’dynamics’ behind this talk of stability appears. A (triangulated) category in itself would not appear to suggest ’movement’. A quick looks confirms my fears that this is going to get tricky fast, but I see there’s an analogy between stability conditions on triangulated categories and meromorphic quadratic differentials on Riemann surfaces, such that stable objects correspond to finite geodesics, as mentioned in Dynamical systems and categories, pp. 2-3.
Mind you, I suppose we might have got used to finding movement in unexpected places when hearing that that most static of objects, the prime number, can be considered as a dynamic entity, a prime geodesic.
I don’t think there is dynamics to be found here, that would be strange.
I have something on dynamics, but first to get the thing in #28 out of the way…
I don’t mean anything more by ’dynamics’ than Kontsevich et al. “We study questions motivated by results in the classical theory of dynamical systems in the context of triangulated and A-infinity categories”, but maybe you’d see this as ’kinematics’, as differentiated at kinematics and dynamics.
Anyway, #28 first.
Ah, now I see what you are pointing me to. Hm, need to think about that…
It’s interesting this idea that it’s through some physical interpretation that one gains the best grip on some phenomenon (“the concept find its meaning”), which might have been studied quite thoroughly by mathematicians. There’s perhaps another case at McKay correspondence of the claim that things making best sense understood in gauge theory (“So that then finally is the relation…”).
You might think that the physicists are tapping into one instance of some general structural relationship which extends well beyond anything realised in the world. Sure they might benefit from having a very concrete way of viewing matters, and one could understand Witten’s advantage in tackling some apparently pure mathematics (such as knot invariants) as resting on this physics know-how. But then it would be surprising that the physics penetrates to the core of the structural relationship.
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