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added to string theory FAQ two new paragraphs:
Prompted by the MO discussion
I wonder if there’s something that could be said during your comparison between model building in general relativity and string theory about falsifiability. Woit on MO is representing you as believing no scientific theory to be falsifiable. This is actually a reasonable position. As you say, what typically happens is the exclusion of parts of the range of possible models. To falsify a whole theory is very difficult. This was precisely the objection raised to Popper’s account of falsificationism. Look at Newtonian gravitation. Uranus misbehaves, so we fiddle about with our assumptions concerning the masses in the solar system, and proceed to find Neptune. Mercury misbehaves, so we do the same (Is there a smaller planet Vulcan, is there a cloud of matter about the sun,… and even, should we alter slightly the parameter 2 in the inverse square law?). Nothing works in such a way that we detect anything new, but we can’t know that we’ve thought of all possibilities that would allow us to retain Newtonian theory. E.g., maybe the cloud of matter is right but our instruments can’t see it yet.
Popper used the prediction of starlight bending as his paradigm case of acting well as a scientist, making bold conjectures and risky predictions. Yet of course had the light bent differently or not at all, there could have been adjustments to the models to accommodate the data. Popper’s thought then is that any such modifications shouldn’t happen in an ad hoc way. Lakatos takes this up in terms of his idea of an unfolding research programmes, according to heuristic principles. A programme should have the heuristic resources to unfold in such a way that it overcomes the obstacles in its path, but in the “spirit of the programme”.
Perhaps then critics of string theory can see that there is ruling out of ranges of parameter space, but they don’t see the ’Neptune’-like phenomenon, where a model modification is made, and independent evidence for that modification is acquired. Or the ’bending starlight’-like phenomenon, where a region of parameter space is selected (here from Mercury observations) and a novel quantitative prediction of bending light then follows, which is later observed.
I think it feels to the critic as though string theory allows reduction of viable parameter space, but has insufficient a strength of heuristic to indicate where to go next. I should think a part of this is a false sense of what kind of speed of progress is to be expected after the revolutionary years of the early twentieth century.
By the way, I know you’re a visionary, but presumably the date in
Loads of string theoretic models and non string-theoretic models have been excluded by LHC data in 2021
should be 2012.
Thanks for that comment. I will have to go to our group seminar now, but I’ll take up this suggestion and add some more comparison to other theories later, such as you are suggesting.
Maybe even stronger than your comparison to Newtonian gravity is the following example:
modern cosmology based on the FRW model in Einstein general relativity is faced with some major Neptune-like phenomena: it only works if one assumes that the majority of all matter and energy in the universe is something that is otherwise completely unknown. But dark energy and dark matter can be added by fiddling with the model.
Then it actually works quite well, and the majority of cosmologists take this as confirmation of the model. On the other hand, some proponents of MOND take it as the opposite and complain that the standard model of cosmology is “unfalsifiable” as no matter what happens, people will fit its parameters, and be it by positing that the vast majority of what the model models is “dark” and unknown. These people (these are few, but they exist) argue that instead Einstein gravity is just not a good theory of reality and needs to be replaced. By MOND.
Whichever position one takes in this issue (if any), it may be worth contemplating as an analog example.
Okay, I have now added something in a new section
Aside: How do physical theories make predictions, anyway?
Maybe it’s all too verbose now. If anyone feels like polishing it, please feel free. I care more about the poitns raised than the way it is formulated.
Have tidied that section up a bit. It’s looking good!
I didn’t quite understand
(but then it is very well so)
By the way, you should never say “despite of”, but either “in spite of” or “despite”.
Have tidied that section up a bit.
Thanks!
It’s looking good!
Okay, thanks, great.
I didn’t quite understand
(but then it is very well so)
Ah, I meant to express the following, have now expanded the line accordingly:
only with this assumption is the model consistent with observations (but with this assumption it is very well consistent with observation)
.
By the way, you should never say “despite of”, but either “in spite of” or “despite”.
Ah, right. I should know this. Thanks for alerting me. Hope to remember that next time.
@Urs: the latest edits (for example, the new Aside) are very well-written, very clear. Thanks for taking the time to do this.
I made a few small changes. The English verb “fix” has at least two meanings: one is “to hold steady” (e.g., hold under consideration), and the other is “to repair”. Both senses were used (in fact, both senses once appeared in the same sentence), so for clarity I either changed to a synonym or added a brief clarifying parenthetical. For example, instead of “fix a theory” I changed to “posit a theory”, and instead of “see if this can be fixed” I changed to “see if this can be repaired”.
I also inserted a few words surrounding the description of Einstein’s “biggest blunder”.
Thanks, Todd!!
One should probably also add some more pointers to sources for that little historical discussion. Maybe I find the time to do so. But if anyone else feels like it, please don’t hesitate.
now filled in text in reply to the question How is string theory related to the theory of gravity?.
prompted by this Physics.SE question
Reading what you wrote there gave me the idea of putting together your audacious speculation
- perturbation/low energy truncation of physics homotopy truncation of sphere spectrum,
with a thought of mine
homotopy truncation … tends to make things more complicated,
Does this give us a clue as to why perturbative treatments of string theory are complicated?
Hi David,
One general comment I’d have is that the perturbation series in string theory is not specifically more complicated than the average explicit computation in perturbative QFT. In fact it is just the sum over a bunch of ordinary QFT correlators. Conceptually that’s very simple. Of course writing out all the terms and dealing correctly with measures on moduli spaces etc. tends to quickly become technically tedious.
But concerning your intuition that the non-pertrubative theory ought to look “simpler” or let’s say “more natural” than its perturbative expansion is certainly an intuition that I would share and that I think other people share, too.
We could wander off into very speculative territory with this thought. But maybe first it’s better to look at some examples where we actually know how this works.
Namely this here is a good example to keep in mind (and I should add it to the string theory FAQ when I find a minute):
the relation between the string perturbation seris and the non-perturbative theory that it approximates is actually understood in the toy example of what is called the “A-model/B-model topological string”.
There are some remarks on this at TCFT – Effective background field theories.
Namely the argument is that the effective background theory of the A-model topological string is nothing but 3d Chern-Simons theory. This was first argued by Witten in 95, and then discussed in more mathematical detail by Costello in 06. But Chern-Simons theory has a known non-perturbative formulation.
Similarly the effective background of the B-model topological string is supposed to be BCOV-theory.
Thanks, I’ll take a look. It would be great if increase in naturalness went along with de-truncation.
added a new section:
(using this Physics.SE reply of mine)
Added three more paragraphs to the beginning of the answer to the question Does string theory predict supersymmetry? .
(This is prompted by some confusion on some notoriously confused blog…)
added a brief section How do strings model massive particles?, in reply to this Physics.SE question
have added a section What does it mean to say that string theory has a “landscape of solutions”? in response to this Physics.SE question
In case anyone has gotten fed up with my string theory FAQ-entries and wants to hear somebody else: Matt Strassler here makes an excellent effort to reply to those frequently (and again and again) asked “questions”.
I wonder who would be getting “fed up”? I think the page is awesome. (But I enjoyed reading the exchange involving Strassler as well.)
Okay, thanks for the feedback!
Wrote some paragraps for a new question
This is in reply to this question on the physics.SE forum.
Prompted by user feedback from Dan Piponi (here in the comment section) I have added to Does string theory predict supersymmetry? after the sentence which says in prose that assuming strings and fermions implies supergravity, the same again as the slogan:
I have added a further item to the string theory FAQ: Why not consider perturbative p-brane scattering for any p?
added another item: What are the equations of string theory?
But perturbative string theory is not a local field theory
Should it then also point out that a non-perturbative version of string theory will satisfy equations, being local?
I see later on the page ’non-perturbative string theory’ links to M-theory.
True, I have added a paragraph on equations of motion in string field theory.
Was there p-brane section there all along? The string theory FAQ article is quite a bit less technical than others, sections like “why is string theory controversial” are quite essay’ish and I’d say even almost educated layman comprehensible. Let me just point out that now the required expertise spikes much earlier.
The string theory FAQ article is quite a bit less technical than others
That’s the point of an FAQ.
Let me just point out that now the required expertise spikes much earlier
An FAQ is not necessarily to be read in sequential order. But if you think it would help potential users, feel invited to reorder the items any way you seem fit.
There’s probably something to extract from Kane’s String/M-theories About Our World Are Testable in the traditional Physics Way.
There’s a frustrating lack of citations for many claims in that paper.
This is an informal review for a philosophy meeting of arguments that Kane and collaborators have been publishing for a while. The main actual research article on the claims in the review is this one here:
This reference and many more are collected at M-theory on G2-manifolds – References – Phenomenology.
I was wondering more about such things as whether his answers to ’Is string theory testable?’ similar to those on this page. And I think generally speaking they are. I.e, the form of testing there is little different from the rest of physics.
Yes, Kane stands out as reminding the public of what ought to be obvious, that and how string phenomenology proceeds just as any other kind of phenomenology, by scanning classes of models in the theory and checking what their assumption predicts. He also highlights, by aiming to give practical examples, that in principle string phenomenology is more constrained, admitting broader conclusions from fewer assumptions.
Still, assumptions remain, as everywhere in model building. Alain Connes with his first prediction quickly contradicted by experiment, and then his re-prediction of the Higgs mass from arguments about spectral triples is a good example of how the prospect of correct phenomenology may be exciting enough to carry even those who should know better across the border of due restraint.
Similarly, Kane has variously been criticized for stating and consecutively re-adjusting definite predictions for gluino masses. Sociologically the process may have been sub-optimal, but as far as I see for each of these predictions the assumptions that it relied on were given, even if not in the press releases. What counts in the end to evaluate any physical theory is to have a good understanding of which models in it produce which predictions, irrespective of the ambient politics. Here it seems to me that Kane stands out as one of the very few who aims his research at investigating this systematically.
As you have probably already seen, the video recording of Kane’s talk is also up.
Maybe noteworthy is the comment starting 21:23. After summarizing the work that Acharya, Witten and other had done up to 2004 as summarized in Acharya-Gukov 04, Kane says:
21:23 “Witten sort of took the attitude: well,the whole thing is set up, we are done, this is the right answer. He went off and did mathematics after that. But this is where we began.”
Notice the issue that is being recalled starting 21:59, that the phenomenology is usefully set up, and consistently so, in the “fluxless sector”, i.e.for vanishing field strength of the supergravity C-field. Incidentlly, that is what drops out from the mathematics when taking the super -cocycles for the M-branes and globalizing them, as I explain in Structure Theory for Higher WZW Terms (schreiber).
So what happens for non-zero flux, which is surely more interesting/relevant? Or is there reason to stick to the fluxless sector other than “that’ all we know how to do”?
David R., it’s the other way round:
flux compactifications are the most studied setup for moduli stabilization (that’s where the “landscape” issue comes from!) but Kane et al. argued in Acharya-Bobkov-Kane-Kumar-Vaman 06 (letter) and Acharya-Bobkov-Kane-Kumar-Shao 07 (details) that for realistic phenomenology, namely to solve the hierarchy problem, fluxless compactifications are necessary (and possible: they argue that moduli are still stabilized by non-perturbative contributions to the potential).
Notice that “fluxless” here means that the bosonic part of the 4-form field strength vanishes, the fermionic part locally with components remains, this is the part that you and me had long discussion related to in Paris. The maths of this “fluxless” sector is not trivial, in particular there may be non-vanishing torsion classes non-rationally, not seen by the field strength.
A priori there is no reason why the theory would pick the fluxless sector, though, as that seems to be a specially fine tuned sector of its configuration space. Kane comments on this in his talk, maybe he is saying that Acharya has shown that this assumption is stable under dynamics, I am not sure at the moment which article he is referring to at that point.
But I find it remarkable that if we take the point of view of higher Cartan geometry, saying that when faced with a -gerbe on a local model space then we are to globalize it over -étale stacks, then this gives just the sector among M-theory on -manifolds that Kane et al. argue to be the realistic one.
there may be non-vanishing torsion classes non-rationally, not seen by the field strength.
Ah, I see – it’s probably a terminology mis-match. What’s the anomaly cancellation condition up in 11d for the C-field? (I see Witten 96 at 11-dimensional supergravity but perhaps this is still an open question and what the cohomotopy is supposed to sort out!) Or is this something else?
I am not sure at the moment which article he is referring to at that point.
this is one part, objectively speaking, where Kane’s accompanying article lets him down: many claims of things that have been done or are known, but with few references.
What’s the anomaly cancellation condition up in 11d for the C-field?
Hopkins-Singer in Quadratic Functions in Geometry, Topology, and M-Theory identified the “flux quantization condition” from Witten 96 as saying that the -field participates in an “integral Wu structure”. For a natural formulation in terms of smooth stacks see section 3.3 of our The moduli stack of the C-field arXiv:1202.2455.
but perhaps this is still an open question and what the cohomotopy is supposed to sort out!
The above is assuming that the -field is in ordinary (albeit twisted/shifted) cohomology. The cohomotopy is meant to be one possibility for refining this further, because fundamentally the -field must be in something richer than ordinary cohomology in order for it to reproduce twisted K-theory after dimensional reduction.
I am not sure at the moment which article he is referring to at this point
On a second thought, possibly its Acharya’s “A Moduli Fixing Mechanism in M theory” (arXiv:hep-th/0212294) which argues that generally the flux is stable.
many claims of things that have been done or are known, but with few references.
In the accompanying article he gives lots of references, but it’s clearly not meant as a technical reference. But let me check with him and I’ll get back to you.
I did receive a reply, but maybe I don’t have the time at the moment to go through the references that I am being given.
But check out the very first paragraph of
I’m reading it, but I don’t get it - and I don’t think it’s the physics lingo. Let me think some more.
Sorry, you say you are not getting what?
That first paragraph is
When the seven-fold possesses a metric of holonomy, then is a vacuum solution of Einstein’s equation. Further, there exists one covariantly constant spinor on , leading to an effective theory with supersymmetry in four dimensions. However, in contrast to M-theory compactifications on Calabi-Yau four-folds [1], if we generalize this background ansatz to allow for non-zero -flux and a warped product metric on , then no supersymmetric vacua away from the trivial background exist. This was demonstrated, for instance, in [2], [3].
So, sure, given a -manifold then we get what I presume is some sort of SUSY-YM on Minkowski space . The logic of the sentence
if we generalize this background ansatz to allow for non-zero -flux and a warped product metric on , then no supersymmetric vacua away from the trivial background exist.
escapes me. Namely they seem to be saying “if we have nonzero and possibly a non-product compactification, we don’t get supersymmetry away from .” Can you clarify? (if you have time) Do we need that supersymmetric vacuum? It also seems like a mangled sentence.
Here “allow for non-zero -flux” means “allow the -flux to take possibly non-vanishing value”.
They are simply saying that in the class of KK-compactifications of 11d SuGra on a manifold that is topologically a product and whose Riemannian metric has the form (“warped compactification”) with no constraint on the -form, then a necessary condition for one Killing spinor to exist (that’s susy in 4d) is that .
By the way, I suppose regarding the “stability” issue mentioned before, this is meant to refer to the superpotential that the rest of Beasley-Witten is concerned with (following Acharya-Spence and Gukov). That superpotential that they discuss (essentially the cup square in differential cohomology) has a minimum (a global minimum, I suppose) for .
I almost got that far, but it was the way the sentence was worded, and the unclear logic of the thing. I suppose we want a Killing spinor to exist? What happens if it doesn’t? Do we pack up and go home, because then there’s no fermions? (even assuming SUSY is further broken so that we don’t have a global supersymmetry, ie what we observe so far).
Or does that minimum at mean that’s really where we should be paying attention, for good physical reason?
Right, the condition that there is a Killing spinor is entirely a choice made by hand, not something required by the theory. It was always motivated just from the phenomenological belief that reality exhibits low energy supersymmetry at the weak scale, and the wish to discuss string compactifications that produce that.
This is discussed at some length in the item Does string theory predict supersymmetry? in the string theory FAQ.
Now that superpotential we are talking about won’t change this situation, I suppose, it may just give a dynamical reason for vanishing -flux. While the claim is that vanishing -flux is necessary for susy, I don’t think it’s sufficient.
That’s as far as the traditional literature is concerned. But maybe it’s not the end of the story. I find it remarkable that while the condition for the existence of a Killing spinor in string compactificaiton was originally motivated entirely by prejudices about real world phenomenology, it just so happens that it is precisely this case for which the mathematics of the resulting compactifications becomes the richest and mathematically most interesting. For instance all of mirror symmetry arises in this case, but not otherwise. For that reason, strings on -compactifications – Calabi-Yau fibration – have long began to play a role of their own in the mathematics literature. The historical reason that mathematicians speak of “Calabi-Yau objects” in monoidal -categories is ultimately because string phenomenologists considered strings on Calabi-Yau manifolds as a promising model for weak-scale supersymmetric phenomenology.
So if one feels at all that the appearance of rich mathematics in a physical theory is not a coincidence, then maybe something deep is still hidden here.
Personally, my best guess is this: as already mentioned in #33, 11d SuGra with M-brane corrections included (and that’s 95% of what is being called “M-theory” in practice) is mathematically exactly the theory obtained by taking the exceptional WZW super-2-gerbe on as the higher Kleinian model space, and then asking for the higher Cartan geometries that it induces.
But now here is an observation: that 2-gerbe need not necessarily be regarded as real
we may also regard it as complex
If we do that, while still requiring that it is a higher pre-quantization of the M2-cocycle, then we are now free to add to its connection 3-form a closed imaginary part
. Now the point of is that is the exceptional super Lie algebra cocycle on , for the left-invariant super-forms. Presently I don’t know any really good mathemathical reason to prefer any choice of over any other, but from the form of this expression it is suggestive (and maybe somebody sees a good abstract reason?) that we take
for the -s ranging through , and for any fixed spinor. For one, this indeed gives a closed form. On the other hand this is the associative 3-form on and hence is the first thing that comes to mind when proving that is indeed a cocycle by using octions.
In any case, that choice of is certainly somehow special, I would be happy to know some principle why we would “have” to pick it.
For if we do pick it, and if we then compute what the higher Cartan geometry is that is obtained by globalizing , then of course we find that this is no longer just general 11d sugra with M-brane corrections, but it is now 11d sugra on -manifold fibrations (for that’s what the globalization of enforces).
I think this is suggestive of something. But even if so, something still seems to be missing in the story.
Finally, an additional observation is that the 2-gerbes that globalize this are special in that their volume holonomy over supersymmetric 3-cycles is precisely the membrane instanton contribution. That also means something.
(This works as follows: supersymmetric 3-cycles are calibrated submanifolds of the -fibers, and on these restricts to the volume form, hence contributes the otherwise missing kinetic term to the M2-instanton. This is a curious way to get kinetic action contributions entirely from a topological theory of WZW terms… )
Thanks, that makes a lot more sense. I suspected that a lot of the phenomenology assumptions I don’t understand come from an a priori assumption of what people expect, and not just what we observe (as opposed to mathematical necessity eg anomaly cancellation). This is reasonable if people are trying to produce models that predict something we haven’t seen yet–due to insufficient accelerator power or data–but still might, and so find reason to prefer one model over another. But it’s putting the cart in front of the horse if one is trying to derive what we should look at, which is the method I understand better.
it’s putting the cart in front of the horse if one is trying to derive what we should look at
While I am the first to share that sentiment, let’s not forget that, while it is easy to demand that fundamental reality be derivable from first principles and pure mathematics, it would be unprecedented and most astonishing.
I should have emphasised the ’if’ in that sentence!
And, we must admit, a lot of what is possible in the higher geometry world is because people were crazy enough to, for instance, try to write down 11-dimensional supergravity, so we (eg John B and John H) could unearth the exceptional higher Lie algebra cocycles.
I was reading earlier today about the apparent controversy over whether Einstein or Hilbert got the GR field equations first (they had it clear in their minds where the credit was due): exactly which day in November 1915 after which letter was received by whom etc etc. But it is clear that one had the physical derivation worked out, even if the right idea was not properly captured in the formalism for a while, and the other had a general mathematical principle and the right input that recovered the same end answer (but without good a physics story). I guess this feels a bit like the same thing: we are in the period 1912-14 perhaps, where the mathematics is still coming to the aid of the physical intuition.
And, we must admit, a lot of what is possible in the higher geometry world is because people were crazy enough to, for instance, try to write down 11-dimensional supergravity, so we (eg John B and John H) could unearth the exceptional higher Lie algebra cocycles.
Hm, no. The cocycles for the M-branes were pointed out by D’Auria and Fré in the same article (’82 pdf) in which they show that 11d supergravity has an elegant formulation in terms of them. The cocyles for the various strings were classified in 87 (here). The fact that this means that sugra is nicely formulated as Cartan connections with values in super -algebras is in my article with Sati and Stasheff, but check out the very first article on the Cafe for the pre-history.
OK, amend that to ’discover the iterated extensions/globalisations of higher Cartan geometry underlying the brane bouquet that includes 11d SUGRA’. I was just grabbing a random example and it was the first that came to mind - I didn’t stop to check the history. Actually, I should have said the Lie-n-algebras, not the cocycles, but anyway. I hope the point is kinda clear that I don’t begrudge furious model building even if it is merely exploring some corner of the parameter space without a skerrick of experimental justification.
merely exploring some corner of the parameter space without a skerrick of experimental justification.
This is maybe a bit subtle. At least for a long time many people were convinced that weak-scale supersymmetry is the best explanation for the available experimental data. Those particle physicists never cared much about the mathematical nicety of supersymmetry, they preferred it because they found that’s what experiment points to. And I think it is at least unwise to follow the suggestion, made elsewhere, that those researchers who still think that’s the case are doing so for general lack of scientific integrity.
In that vein it is interesting to watch the recording of Kane’s talk beyond approximately minute 30, when that debate with the audience starts. I can’t help but find that in this discussion Kane stands out as the only sober participant. The audience that does not want to believe in the long published claim that he recounts should go the usual way of science, and ask him to explain that claim in detail, for them to check. Just throwing out insinuations is not leading anywhere.
ask him to explain that claim in detail
this is why I’d like Kane’s article to be saturated with references. It has lines like ’we know A, B, C, ….’, with merely a sprinkling of references, or at best, poorly sign-posted references, so even someone like yourself is not immediately sure what paper backs up which claim. (see section 9 for the worst offending section). Given a presentation with people skeptical of string theory in the audience (in which camp I would confess to having one foot), this is the chance to gather all the literature together to evaluate potential gaps in the argument.
The claims he reviews have all long been published in the standard hep journals, he showed the arXiv numbers on his slides. Specifically the claim that caused the audience in his talk to lose their temper is reviewed here:
I have received reply from Bobby Acharya now. He kindly confirms that the stability issue in question is indeed that suggested by Acharya-Spence and then discussed in more detail by Beasley-Witten, as in #38, #42 above.
I guess it’s a balance between a paper for those who know the literature and the field, and those who are coming to the issue without knowing such things. Certainly having arXiv numbers on the slides is no use to a reader of the paper a few years hence when the conference website evaporates or moves.
That slide 29 – man, he needs to read some Edward Tufte … and make references for these grand claims - I can find one arXiv reference after the ’basic framework established’ slide, #25, and a reference to ’over 500 pages’ of papers.
I say this not to whinge and complain, but that if this conference was about giving good reasons for people to trust string theory, listing a long bunch of results with no way to even look up said results (other than point to an unspecified list of 20 or so papers totalling over 500 pages, and one specific paper) does not help. He even says ’too important to leave to the string theorists’! But doesn’t give others the option of getting into the gritty details.
Anyway, I think I’ve made my point, and I appreciate you spending the time to clarify and discuss things, Urs.
That slide 29 – man, he needs to read some Edward Tufte … and make references for these grand claims
Every item on that slide is reviewing published material. The references are all listed, one by one, in section 7 of the article:
moduli stabilization: arXiv:0701034
gravitino masses: arXiv:1408.1961
heavy scalars: arXiv:0801.0478
suppressed gaugino masses arXiv:/0606262
solution to hierarchy problem: arXiv:0701034
non-thermal cosmology: arXiv:0804.0863
baryogenesis arXiv:1108.5178
axion stabilization arXiv:1004.5138
Higgs mass: arXiv:1211.2231 (that reference is indeed missing in section 7)
electric dipole moments arXiv:1405.7719
gluino predictions arXiv:1408.1961
Prompted by people wondering about remarks in Roger Penrose’s new book on the (lack of) stability of 4d spacetime in the presence of a KK-compactification (e.g. on PhysicsForums here), I have added to the string theory FAQ an item Do the extra dimensions lead to instability of 4 dimensional spacetime?.
That’s reassuring. We wouldn’t want those compact dimensions to rip open.
Does the relevant potential still allow fluctuations around a value?
By the way, I put in a link on moduli stabilization to vacuum expectation value. The latter redirects to a stub for condensate.
We wouldn’t want those compact dimensions to rip open.
Interestingly, Penroses’ claim (in section 10.3 here) is the opposite: in gravity field theory his singularity theorem generically implies that small compact dimensions quickly collapse to a singularity.
I see room that the arguments in the string theory literature for that this does not happen due to stringy flux fields and non-perturbative effects may be imperfect. But I suppose if so, then it is these arguments that would need discussion, and not a different argument in field theoretic gravity. The instability of Kaluza-Klein-compactifications in field theoretic gravity was of course the reason why KK-theory was abandoned back in the beginning of the 20th century.
Does the relevant potential still allow fluctuations around a value?
Sure, it’s just a potential well.
Thanks for the cross-links with vev-s!
I was hinting that something needed to be added but only if there’s the energy. Doesn’t vev deserve a page of its own?
Doesn’t vev deserve a page of its own?
It does deserve a page of its own, yes.
I have expanded the item What are the equations of string theory?, prompted by the question coming up again on PF, here.
prompted by public demand (on PO here) I have added another item to the string theory FAQ:
But I don’t really have time for this at the moment, so I left it brief, with mainly a quote from the literature doing the job.
I corrected a presumed spelling Grross –> Gross, but it looks like you intended Erler-Gross to be a bibliographic reference which is not yet under References.
Thanks!
Fixed the pointer now. It was trying to point to Erler-Gross at causality, but I had a superflous hash sign in there, which confused the parser.
I have added a further item
and made that also the Idea-section of a stub entry perturbative string theory vacuum.
This deserves to be polished and expanded more. But not tonight.
I was looking to correct typos, but when submitting got told the page is locked and that I’m editing.
Anyway, at some point we need to change
; theoris
and I’m not sure what is being said here:
in the given state that the fields are in, which is in, which is defined thereby
I fixed the first two now, but don’t know what the last is supposed to say.
Thanks!
The “which is in” just needs to be removed. I have fixed it now, also at perturbative string theory vacuum.
I wrote this in a bit of a haste, when somebody asked while I was really occupied with something else. Will try to come back to this. Possibly when finalizing chapter “15. Scattering” of the QFT series.
Looking through some links from that answer, 2dCFT and 2-spectral triple, and on to your Physics Forums article, Spectral Standard Model and String Compactifications, got me wondering about the spectral approach. If the treatment in Modern Physics formalized in Modal Homotopy Type Theory and dcct looks to the geometric side of the algebra-geometry duality, is it that one can mimic all this dually in the algebraic operator approach?
Should there be a native logic for the algebraic operator approach, or are we just too wedded to the spatial?
Bit of a vague thought, but it is a Friday afternoon.
I have a vague idea: Much NCG is secretly higher geometry. For instance most of the C*-algebras considered in Connes-style NCG are groupoid convolution algebras.
Moreover, the Wick algebras of free quantum fields are Moyal star-product algebras. But finite dimensional such are again (polarized) groupoid convolution algebras, namely of the symplectic groupoid (here).
Now the QFT Wick algebras are not finite dimensionaL BUT the relevant infinite-dimensional symplectic structure arises via transgression from the finite-dimensional but higher Poisson bracket on the jet bundle (seee the chapters “Symmetries” and “Observables” in the A first idea..). My hunch is that we may do (higher) groupoid convolution of the (higher) symplectic groupoid up on the jet bundle to get a finite-dimensional non-commutative algebra there and then somehow transgress that down to spacetime to the non-finite dimensional algebra of quantum observables.
This idea needs work. From spring on I will finally have time for research again… But right now I am sort of confident that this will work.
If this works out, then it would no longer be crazy to speculate that this works rather generally. But we have to see.
Interesting! Not sure I’d ever noticed that subsection Higher groupoid convolution algebras.
Roll on spring-time.
I enjoyed the introductory chapter.
Section 3.2 might have earned me a second non-self citation for
But no. All I have so far is in A Schema for Duality, Illustrated by Bosonization:
pure mathematicians sometimes work with uninterpreted theories; and duality is a grand theme in mathematics, just as it is in physics. But although comparing duality in mathematics and in physics would be a very worthwhile project, we set it aside. Cf. Corfield (2017)
Oh, I see, sorry, good point. We haven’t sent back the galley proofs yet. I think. I’ll add in the citation now.
Thanks! Anyway, the later sections were heavily dependent on your input.
All the more should I have cited it. But, to be frank, I had lost sight of its existence. You could have reminded me earlier. But it looks like the citation will make it to the final version now.
In the item “Does string theory predict supersymmetry?”, after the paragraph saying that there is/was no theoretical reason for global compactifications, I have added the following line:
However, recently arguments for a theoretical preference for supersymmetric compactifications have been advanced after all (FSS19, Sec. 3.4, Acharya 19).
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