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Typo:
quaternionc
$S^4 \sslash \widehat{Sp(2)}$ and $S^7 \sslash \widehat{Sp(2)}$, respectively, where $\widehat{Sp(2)} \,:=\, Fivebrane\big( Sp(2)\big)$.
I like this!
Thanks! Fixed now.
The green text in Def 4.2 looks like it might be missing a hat.
True, thanks! Will fix tomorrow.
We are finalizing this followup article:
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This adds proper equivariance to the twistorial character map discussed previously, and proves that upon flux quantization this includes the localization onto the heterotic M5-brane locus:
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Abstract. The Green-Schwarz mechanism has been suggested to secretly be a higher gauge theoretic phenomenon, embodying a higher Bianchi identity for a higher-degree analog of a curvature form of a higher gauge field.
Here we prove that the non-perturbative Hořava-Witten Green-Schwarz mechanism for heterotic line bundles in heterotic M-theory with M5-branes parallel to MO9-planes on A1-singularities is accurately encoded in the higher gauge theory for higher gauge group of the equivariant homotopy type of the $\mathbb{Z}_2$-equivariant $A_\infty$-loop group of twistor space. In this formulation, the flux forms of the heterotic gauge field, the B-field on the M5-brane and of the C-field in the M-theory bulk are all unified into the character image of a single cocycle in equivariant twistorial Cohomotopy theory; and that cocycle enforces the quantization condition on all fluxes: the integrality of the gauge flux, the half-shifted integrality of the C-field flux and the integrality of the dual C-field flux (i.e. of the Page charge in the bulk and of the Hopf-WZ term on the M5-brane). This result is in line with the Hypothesis H that M-brane charges are quantized in J-twisted Cohomotopy theory.
The mathematical heart of our proof is the construction of the equivariant twisted non-abelian character map via an equivariant twisted non-abelian de Rham theorem, which we prove; as well as the computation of the equivariant relative minimal model of the $\mathbb{Z}_2$-equivariant Sp(1)-parametrized twistor fibration. We lay out the relevant background in equivariant rational homotopy theory and explain how this brings about the subtle flux quantization relations in heterotic M-theory.
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Comments are welcome. If you have a look, please grab the latest version of the pdf from behind the above link.
Some quick typos
anaysis; equivarint; co-variantfunctors; equvariant; the the (twice)
Thanks! Fixed now.
Thanks for the alert.
A draft of a possible brief reply is now visible in the Sandbox. Comments are welcome.
Funny these kind of questions. Where else would they expect answers to come from but from authors summarising things already said in their papers. In which case you may as well direct the question to the author and ask for elaboration about what you didn’t get.
On the other hand, it’s a good opportunity to get people to look at your work.
There’s a typo- accomodate.
Good. There really should be pure mathematicians excited by your work.
Hm, does MO not allow to have hyperlink anchors inside a reply?
Have now posted the reply: MO-answer:377154/381.
Thanks again to Dmitri for the alert.
As I commented at MO, the actual question (what can twisted cohomotopy do that other approaches can’t?) is only answered in a nondescript one-sentence paragraph near the end (but before a massive list of references), referring to points made in the previous paragraph in an mildly ambiguous way. I think it a fine explanation, but this should be highlighted in a tl;dr preferably at the top. I skimmed the answer and couldn’t tell where the actual point was. It just looked like a wall of exposition text. It took a close reading to find what I was looking for. Maybe you could say that’s how anyone wanting an answer to the question should read it, but someone unfamiliar with the material is not helped by a small essay if they just want the answer to the question, not the whole background story. That’s not to say the background story shouldn’t be there!
The very first paragraphs of the reply highlight that subquotients of ordinary cohomology groups are not ordinary cohomology anymore – or else you’d conclude from the AHSS that everything is ordinary cohomology. There is a single flux quantization condition enforced by ordinary integral cohomology, namely integrality of the charge, and evidently that’s not the subtle web of conditions in question here. If C-field flux were quantized in ordinary cohomology, there’d be no need for DFM to discuss “models of the C-field”, namely candidate non-ordinary cohomology theories whose structure enforces the peculiar nature of C-field flux. From plain ordinary cohomology in 11d evidently K-theory does not follow, or else the latter were pointless. Ordinary cohomology quantizes non-interacting abelian gauge flux and nothing else, that does’t make an M-theory. It’s extra conditions on top of superficially ordinary cohomology classes which are the content of DMW’s old argument, as explained in the reply’s first section.
One needs to understand, as readers venturing into this discussion might be expected to know, that the Chern-Dold character map and the AHSS are tools to break down any generalized cohomology into ordinary cohomology with a sequence of conditions and identifications imposed on them. It is in this way that, conversely, an intricate web of flux quantization conditions on superficially ordinary cohomlology classes may be unified and be explained by making manifest a single but generalized cohomology theory which enforces them all. It is this nature of flux quantization in generalized cohomology which is the point, cause and aim of the discussion. The reader unclear about this would need to be provided with more explanation of the background story, not less.
It’s not about careful argumentation, but presentation. Winning people over who only have clicked on the question out of curiosity while browsing is not about giving them an essay, but a quick glimpse of why they should slow down and read the detailed justification. Even worse, imagine a string physicist searching for ’hypothesis H’ in Google and finding that question. They would also like to know why they should care. Currently they have to wade past Steenrod operations, the Atiyah–Hirzebruch spectral sequence, the Sullivan model of the 4-sphere. Maybe this doesn’t phase them. Or maybe if they saw at the top of the answer “Hořava-Witten Green-Schwarz mechanism in the presence of M5-branes” they would know this is something they really care about. And then the rest is there so they can invest time in learning what they need to fully appreciate the statement.
Just trying to help you get the message out.
Speaking less politely, for sake of clarity:
It is absurd to suggest that charge quantization in plain ordinary cohomology is of any relevance here. Approached with a question presupposing such a misled notion it behoves any decent reply to start out with explaining the fallacy, in order to arrive at a logical basis on which to lay out a relevant answer.
If a basic clarification of the fundamentals of the topic at hand appears like an inhibitively long “essay” beyond the reading comprehension of the impatient reader looking passively to be “won over”, then that the reader has ventured mistakenly into a subject above their head and should kindly look for leisure reading elsewhere. It is furthermore absurd to suggest that a reader caring to engage in discussion of the topic at hand would be hampered by notions of basic algebraic topology, for if not in these terms there is nothing contentful to be said here.
I heartily disagree with the all too wide-spread attitude of #118 that, allegedly: “It’s not about careful argumentation, but presentation.” On the contrary. Where could we be if not for that awful tendency towards the superficial.
Luckily, it’s a disagreement largely off-topic to this forum here, so I’ll leave it at that and instead look forward to further discussion of the maths/physics content of the topic of the present thread!
allegedly: “It’s not about careful argumentation, but presentation.”
That is not what I’m arguing. I’m saying both are important. I wasn’t claiming anything wrong with the careful argumentation, or that the careful details shouldn’t be there. Just that something might need to grab the eye of someone nonplussed about all the technical details at first. Give them a reason up-front to care, as fast as possible. Maybe someone might get a good impression of your work without wanting to know the details.
But as you say, let’s not flog a dead horse. Your answer is fine, just not how I would have structured it, and that’s not something I should complain about too much!
I was thinking about where a pure mathematician might take particular interest in Hypothesis H. Just as someone working on the enumerative geometry of rational curves on a quintic threefold in the 90s could learn from Candelas et al. on mirror symmetry, might a pure mathematician working on, say, elliptic cohomology benefit from:
we think we see now that there is a natural chromatic character map on twisted Cohomotopy which exhibits the M5-brane partition function as charge-quantized in elliptic cohomology, matching traditional discussion of M5-brane ellitptic genera.
In a curious turn of events, the core of the argument in our
turns out to overlap with a more physics-style but otherwise closely analogous argument in Section 4.1 of:
(While we did cite these authors for a series of articles, we did miss this one. It appeared half a year after our first writeup of the M5 anomaly cancellation argument in sections 2.5 & 4.5 in v1 of arXiv:1904.10207v1, while we were busy splitting it off as the stand-alone preprint that it became two months later.)
So these authors agree that in general the basic component $G_4^{basic}$ (they call it $\gamma_4$) needs to be considered and that one needs some extra assumption on the nature of M-theory to see that it vanishes (their appendix C). Their solution to this problem involves, somewhat implicitly but clearly, the characterization of the tangent structure to sphere-fiber bundles which in our article was encoded in the big diagram at the end:
Namely, on sphere fibrations all stable characteristic classes – notably all polynomials in Pontrjagin classes – are basic. This is what drives the cancellation proof.
In reaction, we have now reworked the discussion in our preprint, by splitting our Section 3 into two parts:
The new Sec. 3.1 gives a general mechanism for cancelling the remaining anomaly term.
The new Sec. 3.2 shows how Hypothesis H implements this mechanism – and uses the occasion to review/explain it all in a bit more detail.
The latest version of this v2 is kept behind the above link.
That’s encouraging, I guess. Have you been in touch with the authors?
Not to discuss politics here. As I said, in reaction we have reworked the writeup as indicated in #122. There is enough interesting mathematical content here to discuss, I’d think!
But don’t feel compelled to get into it. I understand you have other things on your mind. While it’s true that nobody here has entered into discussion of the maths/physics content of this thread much, this doesn’t mean that it has to remain this way, and in my endless optimism I have kept and probably will keep posting updates here. Maybe interested bystanders are silently following the maths content after all. Or will in the future.
I’m lurking and reading the updates :-)
What you call ’politics’ is the subject matter of the philosophy of science, but I can understand not wanting it here.
Some typos
as the anomaly polynomials (’of’ rather than ’as’); tiwsted; coccyle
You have
Atiyah-Penrose twistor fibration $t_{\mathbb{H}}$
but in (22) on the right label both maps are called $h_{\mathbb{C}}$. In any case, why is the first map there labelled ’complex Hopf fibration’ when that’s $S^3 \to S^2$?
Re the final point in #126, is it that you’re calling in (22) that map $h_{\mathbb{C}}: S^7 \to \mathbb{C}P^3$, the complex Hopf fibration, since it corresponds to the map of the fibres over $S^4$?
Oh, I see elsewhere you’re calling it ’7d complex Hopf fibration’.
Thanks for catching the typo in the twistor fibration! Fixed now, in the pdf there.
And, yes, it is fairly standard to call also $S^7 \to \mathbb{C}P^3$ a complex Hopf fibration (after all, it’s given by the same kind of construction, sending real lines to the complex lines that they span, as $S^3 \to S^2$).
Apart from this, the critique of the traditional curvature corrections to the $G_7$-Bianchi identiy is now split off from Remark 6 to its own Remark 7 (p. 7). With a paragraph below it leading over to the need to invoke a more fundamental principle for fixing Bianchi identities (namely cohomological flux quantization).
(Now that we suddenly all agree on the remaining M5-anomaly cancellation argument, this question about the form of the $G_7$-Bianchi identity is the key issue of debate…)
And the other typos I mentioned
as the anomaly polynomials (’of’ rather than ’as’); tiwsted; coccle
Ah, right thanks. Fixed now. Also added some more references on p. 8, and made more explicit the splitting used in the proof of Corollary 21.
For what it’s worth, I have bee experimenting with producing title and abstract for the next (second) installment of: “String and M-Theory: The New Geometry of the 21st Century – II” via NUS Singapore in Dec 2021.
A first version is now here.
It does fit very nicely with Friedman’s schema of the need for a new geometry.
Typos:
With the 21st centrury now well underway,
’century’ and ’under way’, usually two words as an adverb
Thanks, I have fixed the century now.
I did check with Google for “under way” and its first (or rather 0th) hit offers both versions but then gives two examples which both use the version with a single word. Now scrolling further down I see the point is the subject of some discussion. But also it looks like Google is messing this up with whitespace, as it claims that “people also ask” the question “Is it underway or underway?” :-)
It seems to be drifting to both being acceptable. I was taught ’under way’ as a predicate adjective, e.g., ’the conference is under way’, but usage changes.
They seem to have omitted you from the participant list.
Now I have some slides to show, for tomorrow: here.
Slide 20, it’s ’né’ for a male.
Slide 23:
But why coefficients like $P U_{\omega}$?
I recall John Baez wondering:
For some mysterious reason, it looks like K(Z,n)’s are important quantum physics! This is especially interesting because the abstract definition of the K(Z,n)’s has nothing to do with the complex numbers - just the integers. The complex numbers show up on their own accord. So maybe this hints at some explanation of why the complex numbers are important in quantum mechanics.
Why are K(Z,n)’s connected to quantum theory? I don’t really know. But we can get some clues by asking some more specific questions.
Thanks for fixing my French!
Regarding $PU(\mathcal{H})$; while this angle is somewhat orthogonal to the intention in the slides, let’s think about it:
At face value, there seem to be only two known $K(\mathbb{Z},n)$-s which are topological groups related to quantum physics, aren’t there. Not counting in $K(\mathbb{Z},0)$, there is the evident $U(1)$ and then all the mystery, if there is such, about $PU(\mathcal{H}) \simeq K(\mathbb{Z},2)$, is contained in Kuiper’s theorem.
Now two examples is really not yet a pattern, in particular if they are so closely two aspects of the same thing.
But back in the days, André Henriques used to claim to have found the next examples in the list, from 2d CFT in AQFT guise. But in the comments below his MO:a/46634 he says (as of Oct. 2017) that the proof remains open (you may have to click on “see more comments”).
I was searching for that old MO discussion just recently, since the theorem about equivariant classifying spaces that we are about to publish (in a matter of weeks, hopefully) makes me want to have more examples of “natural” topological groups whose homotopy type is an EM-space.
If anyone has any news about André’s old idea, or something similar, let me know. Would be great if it worked.
A small suggestion: instead of “$21^{st}$ geometry”, I think “$21^{st}$C geometry” will fit.
If we are talking, as we are, about anni domini, then the traditional notation is Roman numerals and the proper name for geometry happening from the year 2000 = MM on would be MM-geometry.
;-)
For what it’s worth, there is now a first version of slides for a 90 min talk this Thursday at QMU London, with focus, in the second part, on the chord diagrammatics of quantum $\mathrm{D}6\perp \mathrm{D}8$-states, under Hypothesis H.
A brisk pace for 90 minutes.
Since I happened to land on slide 76, there should be $\pi_k(S^4)$ rather than $\pi_k(S^k)$.
Disregarding overlay, cover/toc slides and the multitude of skipped slides (you need to follow the links to see the flow of the talk through the file), there are actually no more than 20 slides to be shown in all of parts 0-II.
Thanks for catching the typo. Fixed now.
They have it advertised as 14-15 UK time on their site. But the Zoom session begins at 13.30, right?
I guess advertising is an internal affair.
Right, I agreed to give a 30 min “pre-talk”, which is what starts at 13:30. That’s why in total there will be $\sim$ 90 min.
Some typos in ’The character map in equivariant twistorial Cohomotopy’
advencements; eqivariant (5 occasions)
equivarint
Thanks! Fixed now (here).
A referee of “M/F-Theory as $M f$-theory” asked why there are no tangential twists considered in this article, even though the domains seem to be spheres. So we have now added some review and amplification of the nature of compactly supported charge cohomology on pages 6-8 of a revised pdf linked here.
Remark 2.2(a)
3-spheres $S^7$
2.2(e)
Finally, while flat Minkowski spacetimes themselves are homotopically trivial, it is this constraint that charges vanishing at infinity which makes them appear to their charge cohomology theory as effective spheres with non-trivial topology
is ungrammatical. ’Vanish’ rather than ’vanishing’?
to understand the n-sphere as being but the homeomorphism type…
’but’ in the sense of ’merely’ or ’simply’ sounds rather old-fashioned/poetic to my ear.
Thanks, right. Fixed now. Have also improved the figure of the stereographic projection.
This situation of twisted differential cohomology of curved spacetimes can of course be discussed, too (cite character article)
Missing citation
Ah, thanks, that was a bad oversight. Fixed now.
An article
referencing Twisted Cohomotopy implies M-theory anomaly cancellation on 8-manifolds
More subtle questions require working on some generalised differential cohomology theory, of which there are many types. The generalised cohomology theory appropriate for M-theory has been postulated in [68]. It would be interesting to see if this more refined picture leads to any interesting consequences in field theory.