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.

]]>Ah, now I see what you are pointing me to. Hm, need to think about that…

]]>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.

]]>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 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.

]]>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. )

]]>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.

]]>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.

]]>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.

]]>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?

]]>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.

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?

]]>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

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.

]]>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.

]]>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:

]]>started rewriting and expanding the Idea-section. Not done yet, but need to interrupt now.

]]>Yes. Thanks, fixed now.

]]>I believe that ’spring’ is an aide-mémoire for Urs to return, as explained here.

]]>Typos

such that for all non-zero objects $E \subset \mathcal{A}$,

and

]]>Then the connected component of the space spring of stability conditions

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.

]]>Put in links for the second part of the opening sentence to help with ’Mumford’s stability’.

]]>added statement of Schur’s lemma for Bridgeland-stable objects, here.

This implies that the answer to my question in #10 is: Yes!

]]>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.

??

]]>