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Urs Schreiber wrote a lot about Hypothesis H on his personal web, maybe the material should be transferred over?
In p.53 of Introduction to Hypothesis H current notes, it says “the Pontrjagin theorem and its variants give, under Hypothesis H, a detailed description of worldvolumes of M-branes as (cobordism classes of normally framed) sub-manifolds of spacetime.”. This still has no information (say, of supersymmetry) of the kind appearing in the brane bouquet, right? So that the alleged non-supersymmetric branes of the kind appearing in e.g. 2305.01012 should appear here if indeed existing?
Both the argument about charge quantization of D-branes in K-theory as well as of M-branes in Cohomotopy applies to possibly non-supersymmetric branes, yes.
For comparison (of either) with arXiv:2305.01012 it would be helpful to extract a table of which charge groups that article means to predict.
For Part 2 (D-branes in Type IIA/B) I gather (from the first full paragraph on p. 13) the conclusion is that the usual K-theory classification is reproduced by the worldsheet argument (all only in flat Minkowski spacetime!).
For Part 3 it requires more work to extract the conclusions drawn.
I see, makes sense.
Could you also comment on some of the consequences of footnote 21 in p.49: ” In fact, together with the canonical coproduct in homology the Pontrjagin product (75) becomes a Hopf algebra structure with the star-involution (76) being a Hopf antipode ” ? Is this accidental or is this somehow expected from the fact that Hopf algebras may be regarded as 3-vector spaces ? There isn’t any restriction on the dimension of the field theory so I can’t really see why one would get precisely a 3-vector space.
That’s an interesting question.
The way it’s conceptualized in the notes, the Hopf algebra is an algebraic reflection of the group structure on loops and as such it seems accidental that it can also be regarded as a 3-vector space.
On the other hand, given the relation of the discussion to the matrix model, it’s maybe noteworthy that in an extended quantization of the membrane one should expect to find a 3-vector space of states.
So maybe it’s more than an accident. I’d need to think about this.
Typo
observabes (p. 47)
Thanks. I have fixed it locally. Will re-upload later.
Is there anything we can say about how these states described by configuration spaces relate to the states classified by derived categories of coherent sheaves in the B-twist in ST?
On p. 21 of Anyonic defect branes we speak of
“Hypothesis D” for the old proposal that D-brane charge is in a derived category with Bridgeland stability, and of
“Hypothesis K” for the less old proposal that it is in K-theory.
Even before it comes to “Hypothesis H”, one would like to know how Hypotheses “D” and “K” actually relate, and (maybe surprisingly) there is essentially no discussion of this point available.
The only comment in this direction we are aware of (cited on that p. 21) is by Eric Sharpe 1999, who points out that for the comparison to even get started one surely needs to look at a holomorphic version of topological K-theory.
I think we add to this the comment that furthermore it must be some form of holomorphic differential K-theory, in order for it to see geometric information such as the actual position of D0- branes which the derived category does see but purely topological K-theory certainly does not.
With this one is led to guess that for comparison with the old “Hypothesis D” one should really be looking at K-theory in its incarnation as a Hodge-filtered differential cohomology theory.
That remains to be worked out. First we thought that our analysis in “Anyonic defect branes” supports this point, but (while it certainly does not contradict it) one cannot quite tell yet because these configuration spaces in 2d are Stein domains so that the difference between the plain and the holomorphic de Rham complexes is not actually visible in this case.
I have added pointer to the lecture notes
the Part 1 of which is now in fairly good shape.
(Comments are welcome. I am giving two further presentations next week, one to the String Theory group of Yau Math Center at Tsinghua, Beijing, the other to the Wolfram Science Winter School, online. Notice that, most unfortunately, I had to cancel my lecture series at Srni Winter School later this month, because of visa renewal deadline that I had completely lost sight of this year.)
This looks very interesting. Could I press you on the motivating logic, which I’ll try to reproduce with some questions in brackets?
The primary problem in physics is a lack of a non-perturbative QFT. We don’t have one even to account for ordinary matter, let alone quantum gravity. (Is there a toy non-perturbative QFT in some unrealistic dimension?) Since string theory was devised to capture bound hadron states, and (is sufficiently promising here?) it’s a good idea to go looking for a non-perturbative version of it, and this is called M-theory. We have an account of what this M-theory is. (Without needing to be so, this M-theory is also a theory of quantum gravity?) (Is it that non-perturbative QFTs are likely to be extremely rare, and realistic ones even rarer (maybe unique)?)
All right, this is all about the motivation on p. 3, of M-theory as an approach to the mass gap problem.
I’ll reply item-by-item:
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The primary problem in physics is a lack of a non-perturbative QFT. We don’t have one even to account for ordinary matter, let alone quantum gravity.
Yes!
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Is there a toy non-perturbative QFT in some unrealistic dimension?
Yes, I have now expanded on this a little more, in this paragraph at non-perturbative QFT.
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Since string theory was devised to capture bound hadron states.
Yes! Originally and now again, decades later (recalled here).
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and is sufficiently promising here?
Yes, that’s the discussion at AdS/QCD correspondence.
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it’s a good idea to go looking for a non-perturbative version of it, and this is called M-theory.
Yes!
$\,$
Without needing to be so, this M-theory is also a theory of quantum gravity?
If I understand you well at this point, and you are asking:
Does it so happen – even though nobody initially asked for this – that the description of QCD via quantized flux tubes happens to be a theory of quantum gravity in an unobserved higher dimensional bulk spacetime?
then the answer is a resounding:
Yes! (e.g. Polyakov 1998)
That this is both confusing as well as intriguing has been the curse and blessing of human interaction with string theory. A process of the Weltgeist slapping itself on the forehead in slow motion, spanning decades (recalled here).
$\,$
Is it that non-perturbative QFTs are likely to be extremely rare, and realistic ones even rarer (maybe unique)?
This depends on which of two commonly conflated meanings of “physical theory” you mean.
There is the map and the territory, and theoretical physicists often speak conflating the two.
As maps, non-perturbative QFTs have remained rare: Barely a satisfactory one has been brought to paper.
As territory, non-perturbative QFT is everywhere: All physics around us is non-perturbative.
The idea of perturbation theory — that Planck’s constant and coupling constants could be literally “infinitesimal” — is a theoretical/intellectual artifice that has no meaning in the real world.
This contrast shows how glaring the problem of formulating non-perturbative QFT is; I had tried to visualize this by the graphics here.
Great, thanks!
As maps, non-perturbative QFTs have remained rare: Barely a satisfactory one has been brought to paper.
As territory, non-perturbative QFT is everywhere: All physics around us is non-perturbative.
Probably more suitable for fireside chat than doing useful work, but … what if there’s a reason for this discrepancy? Something like …
As territory, there’s only the one collection of interacting fields we are faced with. All physics we confront is produced by these fields, even if for practical purposes we separate them out and study them in particular regimes.
While this totality can be mapped by a non-perturbative QFT, there’s no reason that any approximations to it devised for practical purposes need to retain the non-perturbative quality.
Then what if, just speaking structurally, there aren’t many non-perturbative QFTs to be found? There’s so much intricate mathematics deriving from Hypothesis H, it doesn’t seem to be a single general point in a vast array of possibilities. Might reliance on initial constructs such as cohomotopy (as on p. 3) point to its special place?
By the time you’ve incorporated dynamical (super-)gravity into Hypothesis H (p. 57), what kind of latitude is there for setting coupling constants, etc.?
While this totality can be mapped by a non-perturbative QFT, there’s no reason that any approximations to it devised for practical purposes need to retain the non-perturbative quality.
Interesting that you say this. Yes, that’s a thought I had been voicing privately:
Much as it might seem to be so, there is actually no real experimental nor logical reason that fully-fledged “quantization” should necessarily be a general operation on all classical theories, instead of a peculiarity applicable to a few or maybe the single fundamental theory.
On the other hand, there is also a historical quirk in the history of the development of QFT which has skewed the attitude of the field at large: It’s the phenomenal success of Schwinger-Tomonaga-Feynman-Dyson’s perturbative quantum electrodynamics (QED) that led to the abandonment of the Hamiltonian method which the previous generation had established for quantum mechanics.
For reasons that are not even so clear (something to do with $1/137$ being a “small number”, but that’s hardly a convincing explanation), in QED the black magic of taking the asymptotic series produced by pQFT and arbitrarily truncating it after the first few terms happens to match experiment to fantastic precision. When popular texts brag about how “QFT is the most precise theory that mankind ever invented” they are referring to this (ultimately inexplicable) success of pQFT in QED and in QED only.
In quantum chromodynamics (QCD) the situation is completely different (except at ultra high energies). Here it was eventually understood that one needs instead the Hamiltonian formulation, after all, in the guise of light-cone quantization. Had this not remained a comparatively niche approach in the broader QFT community, maybe the development of the lightcone quantization of string/M-theory would be much further developed by now than it is.
By the time you’ve incorporated dynamical (super-)gravity into Hypothesis H (p. 57), what kind of latitude is there for setting coupling constants, etc.?
That’s something far ahead that I cannot seriously answer at this point. Judging from how things work so far, I don’t see that free parameters could crop up, but that’s really speculation at this point.
this (ultimately inexplicable) success of pQFT in QED and in QED only.
I guess another useful check on a future physics that it be able to explain the partial successes and failures of earlier theories, a view I was endorsing back here.
what kind of latitude is there for setting coupling constants, etc.?
That’s something far ahead that I cannot seriously answer at this point.
Might that diagram at the top of p. 3 turn out to be a little misleading then? Or is this just to fall for the Keplerian dream?:
Part of the interest in the idea of the multiverse derives from the feeling that this scenario of the standard model of particle physics, with its particular field content and coupling constants, being fixed by pure logic, is no more likely than Johannes Kepler‘s attempt to derive the distances in the solar system from the Platonic solids (the old Music of the Spheres), which today we understand is a confusion of what is fundamental law and what is random initial conditions. (multiverse)
When I said black magic in #16, I really meant it. It’s worth thinking about how disconcerting the state of affairs in pQFT really is: It produces a series that provably approximates every number between 0 and $\infty$ depending on where you truncate it, without there being a rule for where to truncate it. Now people truncate it to where it about agrees with experiment. If that’s close enough, they declare success, otherwise they fire up a computer simulation instead. That’s pragmatism, but it is not, if one really thinks about it, satisfactory science.
I guess you’re suggesting that it’s not right to think that we have QED sorted (via perturbative methods) and then are only forced to look for a nonperturbative account to deal with QCD topics. Can’t we hope that just as a proper M-theory should provide a small-N corrected holographic QCD (AdS-QCD correspondence), that it should also give us a proper account of QED?
What would one say about how string theory/M-theory can be thought to treat QED? Might Hypothesis H have something important to say there?
Sure, these sectors of the standard model are not actually in isolation.
We saw structure that looks like meson-photon interactions in the M5-brane model emerging from Hypothesis H, here. This clearly needs more work, but the whole aim is to eventually speak about real world physics.
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