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added References to BDR 2-vector bundle .
By the way, whatever happened to the first n-Café millennium prize? Any claimants?
No claimants for the classifying space. But at least the idea that I proposed there, that the right notion is $2Vect = Algebras+Bimodules$ is now being promoted by others, too.
To answer that nCafé-millenium question one will probably have to “put a suitable topology” on these 2-vector spaces, i.e. make good sense of this for the algebras being von Neumann algebras. Christoph Wockel keeps pushing me to write up with him the stuff about the representation of the String-2-group in these “topological 2-vector spaces” that I indicated here. But I keep not having the time. In fact, I badly broke my promise to visit Hamburg for this purpose two months ago. I just couldn’t find the time, after all.
Jack Morava mentioned this paper here
I’ll add it to each author’s page.
Thanks! I have added a more stable link to the article: dml:45387.
Just to add that the string theoretic interpretation of this monopole charge measured by $\pi_3$ of any candidate approximation to elliptic cohomology is the content of M/F-Theory as Mf-Theory. Maybe we should add a remark comparing to BDR’s discussion there.
Is there a particular section of M/F-Theory as Mf-Theory to point to?
p. 3 and then p.12?
Sorry, I mean to be pointing to the interpetation of $\mathbb{Z} \subset \pi_7(S^4) \to \pi_3(\mathbb{S}) \simeq \mathbb{Z}/24$ as reflecting, under Hypothesis H, the folklore result that in M-theory on K3 must have M-branes appearing in multiples of 24.
In the introduction this is p. 16, then in the conclusion it’s p. 86 with an index of further cross-relations on p. 85.
On p. 83 is a remark (turned into a footnote) on how this reflects, as far as one can say this, an argument by McNamara & Vafa.
Does that particular instance of the Hurewicz-Boardman homomorphism show up
$\pi^S_3 \cong \mathbb{Z}_{24} \to K^{alg}_3(\mathbb{Z}) \cong \mathbb{Z}_{48}?$That it’s injective is supposedly explained by Ausoni, Dundas and Rognes.
No, it’s supposed to be just 24.
I wasn’t actively aware, before you now pointed at it, that BDR’s “form of elliptic cohomology” deviates from tmf already in degree 3.
I find it a remarkable coincidence that stable 4-Cohomotopy and 4-tmf are indistinguishable on 10-manifolds, and that this is the maximal dimension for which this is so (Ex. 4.16 on this p. 61).
What’s the “No” to there?
ADR (and so Morava) are wrong to claim (p.797) that
$K_3(\mathbb{Z}) \cong \mathbb{Z}/48 ?$You asked if it “shows up”, and I took that as referring to “in charge quantization of brane charges”. And I am thinking: No, if we’d charge quantize in BDR theory instead of 4-cohomotopy, the story of brane charges seen in $\pi_3$ breaks, by the extent that $\pi_3$ is now larger than $\mathbb{Z}/24$.
On the other hand, the full match to brane charge quantization in 4-Cohomotopy really requires the unstable theory, with its stabilization corresponding to boundary-to-bulk transition. So maybe with some unstable pre-image of BDR it could still work.
I see. Thanks!
In response to my reporting your #8, Jack Morava writes:
Thanks very much for posting this link!
It’s a beautiful and fascinating fact [?] that a twenty-four times punctured compact homotopy K3 four-manifold (arguably a quaternionic analog of an elliptic curve in complex geometry) has a stably trivializable tangent bundle. I’m only peripherally a member of the differential topology community but I believe it is not widely understood there that this has a brane-theoretic interpretation.
I’m only guessing, but I suspect this may be because the mathematical literature around branes centers around questions of mirror symmetry (complex versus symplectic geometry, derived categories of coherent sheaves, stuff like that) which seems to me pretty disjoint from the language used closer to physics (eg ADS/CFT ?). Sorting things out around this particular beautiful example could possibly be extremely helpful.
The trouble with focusing on homological mirror symmetry (in the sense of equivalence of derived/$A_\infty$-categories associated with CYs) is that the interpretation of these derived categories as categories of D-branes in string theory remains based on particularly vague and hand-wavy folklore arguments:
What is fairly well established is that these derived/$A_\infty$ categories encode the branes of the “topological string” (that’s due to an argument by Konsevich which was seminally elaborated on by Costello 2004). But to get from there to the physical string of string theory one needs to identify Bridgeland stability conditions with actual stability of physical D-branes. The state of the art of the argument for that seems not to have progressed since the original articles of Douglas and Aspinwall (here). With all due respect to these authors and their vision, back then, I believe that anyone who reads these articles closely will find that it is fair to say that there are large leaps of faith driven by much handwaving in there.
The section Bridgeland stability condition – As stability of BPS D-branes in the nLab entry on Bridgeland stability is my rendering of what I think, after having spent some time with his articles, is what Aspinwall’s intuition was getting at, back then. I find this a cute argument (at least the way I tried to lay this out in the entry – it is not so immediate to extract this clear picture from the old articles), but cute as the story may be, anyone will agree that this is a really vague and arguably simplistic physics story, with just the most handwaving relation to any first-principles analysis of D-brane stability by what would be their defining worldsheet CFT derivation in perturbative string theory, let alone any enhancement of that to the elusive non-perturbative string theory that is expected to be necessary to really get to the bottom of any of the nature of branes.
Also notice that another part of the community, of people who did worry more explicitly about physical stability of D-branes, ended up adopting another folklore altogether, namely that of D-brane charge quantization in K-theory. This, too, remains a handwavy conjecture with various indications that it’s not completely correct (here), but at least here people did make concrete worldsheet CFT computations in simple examples in order to check if this first-principles derivation from the axioms of (perturbative!) string theory matches the conjectured K-theory classification (upshot: sometimes it seems that it does).
So now there are two traditional proposals: the old one says that D-brane charge is in Bridgeland-stable subcategories of derived categories of the compactfication space, and a less old one says that it is in the (twisted equivariant differential) K-theory of that compatification space.
There is essentially no existing discussion of the relation of these two proposals. What one does see from time to time is an author claiming that “derived categories are better/richer/cooler than K-theory” or that “K-theory is better/richer/cooler than derived categories”.
But all this is moot until there is an actual theoretical foundation of what “D-brane” really means, in non-perturbative string theory, which is where their actual home must be. This is the glaring gap in all contemporary string theory ever since its “second revolution”: It’s meant to be all about non-perturbative effects of branes, but there is no actual theory of these, just a web (vast and suggestive as it may be) of more or less vague plausibility arguments.
The hypothesis that instead of K-theory one must use Cobordism cohomology (ultimately unstable and framed) is of course new, this is Hypothesis H, though the relevance of Cobordism and its plausible relation to 24 brane punctures in K3-compactifications of M/F-theory also appears in the recent field of “Swampland studies” – meaning: while we still don’t know what (non-perturbative) string-theory really is or implies, let’s turn it around and try to identify the complementary, that which is definitely not string theory) – around Vafa, as referenced on that p. 83.
Worth sharing these thoughts more broadly, I’d say.
Have forwarded the reply to the nCafe, if that’s what you mean: now here.
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