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We are finalizing an article
translating concepts in M-theory on flat and fluxless backgrounds to fundamental notions in stable homotopy, generalized cohomology and specifically cobordism theory (assuming Hypothesis H).
Comments are welcome! If you do have a look, please grab the latest version from behind the above link.
Look forward to reading it.
the most fundamental of all cohomology theories is Borsuk-Spanier Cohomotopy theory
I seem to recall asking if there is a way to say this via some universal property, just as in the stable case, one can speak of the initial multiplicative cohomology theory. Perhaps that it stabilises to the initial multiplicative cohomology theory is enough.
Typos from a quick look through the first few pages:
continous; attemtps; cohomoloy
How much more of this magic do you have up your sleeves?!
dicussed; cocofiber; cohmotopy
and I guess that
’co-descent’ should be ’co-descend’.
Interesting feature this idea that
much of this classical material seems to not have received modernized discussion before
So pure algebraic topologists ought to care. Presumably they can read these parts even without the physics inspiration.
Thanks! Am fixing this now.
Yes, the algebraic topology here is mostly from the 1960s. (In the previous articles in the series, the AT was mostly from the 1950s. This is a trend :-)
One could make the point that, historically, differential geometry, as a field in pure mathematics, came into its own only after its application to gravity was understood. Didn’t it?
Similarly, if a field is advertized as being all about computing $\mathbb{S}_\bullet$, while nobody has a notion of what this is good for, then maybe the field’s “meaning explanation”, to borrow or abuse a term, has been lacking.
Novicov; Co-ordism
Very intricate the interplay between mathematics and physics historically, but that the latter can hugely reshape and reinvigorate theories in the former is clear.
Thanks again, all fixed now, here. (Also, the table of cohomology theories on p. 20 now has a more complete list references.)
You have $\mathbb{R}^{b-p} \times S^{9-b}$ in (22), but incorrectly $\mathbb{R}^{p-b} \times S^{9-b}$ twice in the table of p. 84.
Thanks for careful reading; have fixed this now. Also fixed the spacing in the right column.
My comment was flippantly put, but I’m serious. What sort of scope do we have here? Can you foresee the point in the near or medium-term future where you will run out of things in the M-theory literature to explain with Hypothesis H? The question is, I guess, how long you can play this game and then move on to the next phase (as it were)? And, what do you think that might be?
Dunno, what could one possibly do with a TOE?
A TOE that unifies foundations of physics with homotopy-type foundations of mathematics?
If anyone has an idea what this might be good for, let me know!
;-)
(I’ll try a serious reply below. But I admit that the question puzzles me. Is it really unclear where a potential discovery of “M-theory” would be headed? I tend to expect that people regard “Hypothesis H” as presumptuous. A prevailing attitude seems to be that this must not be contemplated outside that IAS office. But if your question is at all representative, then maybe the ambition remains unclear?)
You see, the beauty of the M-theory conjecture is that it reduces a zoo of fields to just two:
Gravity and the C-field, nothing else.
This suggests that, if M-theory really exists, it’s nature is all in that of the C-field coupled to gravity.
But if the assumption is correct that any field species in physics is identified as a
twisted
equivarant
differential
generalized non-abelian cohomology theory – the flux quantization law of the field, then…
…this suggests that identifying M-theory is tantamount to identifying the ted-cohomology theory that flux-quantizes the C-field.
(NB.: It’s the t-e-d adjectives that encode the coupling to gravity.)
If that is the case, we can just guess such a cohomology theory and check – now by rigorous derivation – if it matches the expected phenomena. To the extent that it does, we gain confidence in the guess; to the extent that it does not we try to adjust fine-print (or else, eventually, discard the guess and either try another guess – or retire and instead do Macramé).
In either case, the comparison to “the literature” is just a first-order plausibility check that the guess is on the right track, assuming that “the literature” is. The actual point is to go beyond “the literature” by having an actual definition of the C-field in M-theory, hence, presumeably, of M-theory.
Okay, but maybe you worry about the adjective “quantum” that needs to be brought in. We don’t know for sure yet, of course, but:
The observations in arXiv:1912.10425 suggest that the “higher observables” on (i.e. the cohomology of) the space of C-field configurations, according to Hypothesis H, already are the quantum observables. We began to formalize this statement last year (see this note) but then got distracted by other tasks that seemed more urgent.
So that’s the picture at the horizon, at the moment:
The ted-cohomology of the cocycle space of ted-Cohomotopy on all admissible super-manifolds, that’s the algebra of quantum observables of M-theory, and so is M-theory.
Still some way to go to unravel this. But it’s progressing.
There are suggestions of monumental consequences of correctly formulating M-theory. E.g., at AdS/CFT correspondence it says
While the correspondence is thought to hold also in the small N limit, here the classical super-gravity-computations on the dual (AdS) side will receive small-N corrections (highlighted for holographic QCD e.g. in Sugimoto 16, see references below) from perturbative string theory (for small ’t Hooft coupling) which are hard to compute, and then from M-theory (for small $N_c$) which are largely unknown, as formulating M-theory remains an open problem.
Or of generating a theory of coincident M5-branes, and hence the D=6 N=(2,0) SCFT – Theory $\mathcal{X}$.
Is it reasonable to hope that Hypothesis H will shed some light on either of these in the near future, if it’s correct?
Right, these “monumental consequences” are nothing but the non-perturbative understanding of QCD – the last of the “Millennium” Problems.
String theory promises to give a working definition of non-perturbative QCD, hence of the standard model, by understanding it as the worldvolume theory on a brane, either by working in the bulk spacetime around that brane (AdS/QCD) or by working right on that brane (Theory X)
In all the “quantum gravity” buzz it’s often forgotten that making sense of QCD is the origin, the driving force, and the future of string theory (The wall and the cave).
And indeed, this was the opening motivation for Hypothesis H that I offered in my talk at MTheoryMath2020. All we are trying to do is collect the million bucks from the CMI…
What I lack is any proper sense of how much closer Hypothesis H brings us to this goal, in the sense of a set of steps that need to be traversed.
The journey only just began, but now there is a path ahead.
For the impatient, there is appearance of hadron physics, from Hypothesis H, sketched out in arXiv:2006.00012.
I understand the big picture, I have been paying attention :-)
But I was more interested in the medium-level view moving past this:
In either case, the comparison to “the literature” is just a first-order plausibility check that the guess is on the right track, assuming that “the literature” is. The actual point is to go beyond “the literature” by having an actual definition of the C-field in M-theory, hence, presumeably, of M-theory.
to “second-order checks”
Okay, but maybe you worry about the adjective “quantum” that needs to be brought in. We don’t know for sure yet, of course, but:
I hadn’t thought of this, but yes, that is definitely super-important
The observations in arXiv:1912.10425 suggest that the “higher observables” on (i.e. the cohomology of) the space of C-field configurations, according to Hypothesis H, already are the quantum observables. We began to formalize this statement last year (see this note) but then got distracted by other tasks that seemed more urgent.
OK, so one has to go from just doing all the plausibility checks, to proving new things, uncover new phenomena or whatever. I was wondering if you had a feel for how far you are from starting this phase.
So that’s the picture at the horizon, at the moment:
The ted-cohomology of the cocycle space of ted-Cohomotopy on all admissible super-manifolds, that’s the algebra of quantum observables of M-theory, and so is M-theory.
Excellent, a concrete, mathematical goal. Presumably, aside from the lofty goal of saying “Here Is The Definition Of M-Theory!”, this is what a mathematician or even a mathematical physicist still on the fence about M-theory could look forward to.
That $\subset$ sign in the bottom left of (14) should presumably be the other way around.
proving new things, uncover new phenomena or whatever. I was wondering if you had a feel for how far you are from starting this phase.
We have been pointing out various new phenomena as we proceed. For example:
the $String^{c_2}$-structure on M5-branes in arXiv:2002.11093 is strictly new, even if there have previously been vague speculations in roughly this direction;
you wrote a followup article (arXiv:2003.09832) deriving some more concrete new predictions of this result!
the Hopf-WZ/Page charge quantization in arXiv:1906.07417 is new: there was no general argument or even discussion for that before;
the exclusion of irrational fractional D-brane charge arXiv:1812.09679 is new, providing the resolution of a “paradox” that had remained open in the literature;
the super-exceptional M5-brane model arxiv:1908.00042 is new and predicts a rich tower of hadron-like excitations that we began to chart out in arXiv:2006.00012.
all the rationally invisible torsion brane charges predicted by Cohomotopy are new, such as the non-purple entries in (17) of arXiv:2103.01877;
time permitting, we may next focus on the $\mathbb{Z}_2 = \mathbb{S}_2$ which, by Pontrjagin’s (in)famous result and the discussion around that (17), will describe M5-branes wrapped on tori…
worlvolume (twice) in (18)
Thanks! And thanks for #18. Both fixed now.
OK, I wasn’t aware that completely new results with no precursors/hints/arguments were already coming out!
falt; casses; the this
Thanks again! Fixed now.
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