## M-theory phenomenology - homework exercise

In the current long thread discussing string theory, arivero wrote:
 Quote by arivero it is well known (or well neglected) that in the experimental spectrum there are 84 almost massless fermionic states. They should be protected by some symmetry. Opening at random the Slansky report, I can see 84 in SU(4) (with triality!), SU(6), SO(9),... and I could also look for 42 (hattip Douglas Adams) or 21. So it does not seem a big clue. But the source of the 2-brane of M-theory is the antisymmetric tensor of 84 components, the complement of the 11D graviton (44+84=128) in the N=1 sugra fundamental multiplet. Thus I'd say that the M-theory brane is a candidate to protect the Yukawa couplings of the fermions, in some yet unknown parametrisation of a yet unknown compactification.
I replied
 Quote by mitchell porter Speaking of new threads, I'm going to start one for Alejandro Rivero's idea in comment #425. I don't think the number of degrees of freedom in 11 dimensions is much of a clue for phenomenology, because moving to lower dimensions creates so many new states and relationships. But it would be a good exercise for interested parties to really think this through, and the technicalities might interfere with the discussion here.
So here is the promised thread. As I said, I think it's unlikely that this number from the 11-dimensional theory would show up so directly in 4 dimensions. But it would be good practice to explore the issue more thoroughly. For example, does the proposed mapping even make sense?
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 Blog Entries: 6 Recognitions: Gold Member I do not hope to get too much from the exersice, but lets try! Also, the question about the increasing multiplicity of U dualities could stay in the main thread. My thesis advisor, during 1990-95, was LJ Boya (SPIRES,Arxiv). At that time he was an strong advocate of String Theory, while I had heard lectures from Gracia-Bondia, Coquereaux and Connes, being militant in NCG. Plus, I was hooked on computers, so we just implicitly agreed in a soft topic for the PhD. He gave me a lot of freedom for my own pursuits so I did not followed the developments of "stringers" except from conference to conference. I mention this not only to fulfill point #10 of Baez index (to point out I have been in the school), but also because the first time I heard that the MSSM multiplet was of the same size that gravity, 128-128, was in a very recent observation of Boya, http://arxiv.org/abs/0808.3667 Of course this is true only with massive neutrinos, so nobody was to suggest it in the XXth century, when the neutrino was considered massless. Moreover, if you put the graviton, there is no place for the MSSM higgs (which gives mass not only to the W and Z but also to the quarks and fermions. The point is that with massless neutrinos, the only way to do a 84 out of the standard model was to put the neutrinos in a separate bag and the top in the same status that the rest of quarks. In the XXth century, the number of states to be protected was meaningless, 78 or 72, depending if the protection was to include the neutrinos or not. But with massive neutrinos we have an alternative way with the 12 neutrino states, having a Dirac mass, inside of the 84 and the top in a separate status.
 The first problem is that the 84-component object from supergravity is bosonic. So we would seem to need something which picks out 84 of the 128 fermionic degrees of freedom, and pairs them supersymmetrically with it. The most interesting lead I have so far is hep-th/0703262 and references 3 to 7 therein, which are about extending the "E10" approach to M-theory to fermions. In particular, reference 7 is full of "84+84"s and "28+56"s. But it might take a while to assimilate.

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## M-theory phenomenology - homework exercise

Indeed it seems that the fermions in M-theory are not easy to grasp. Here for instance this quote of Schwarz in hep-th/9706197v1 (underline is mine)
 Quote by John H. Schwarz The massless ﬁelds of M theory are just those of 11-dimensional supergravity: the metric tensor, the gravitino, and a three form potential A3. By the reasoning explained above, there are two kinds of BPS p-branes that can couple to the three-form potential. The one that couples electically is the M2-brane, originally called the supermembrane. Its world volume theory was constructed ten years ago [11]. The brane that couples magnetically to A3 has ﬁve spatial dimensions and is called the M5-brane. Its world volume theory, which was constructed very recently will be described below. The description of the fermionic degrees of the M5-brane involves a number of technical issues that I do not have time to get into here. So I will only describe the bosonic truncation of the M5-brane action. I should emphasize, however, that the complete action with global 11d supersymmetry and local kappa symmetry on the world sheet has been constructed
Two approaches are, in principle, possible: either to extract, in 11D, a piece of the 128 gravitino, by applying the only susy generator to A3 (which is the 84 dim bosonic object), or to go first down to 4D keeping track of all the states produced from A3, and apply the susy transformation there.

(Actually, I suspect that there are two candidate arrangements: the one that started the thread, where the top quark is the piece left inside the 44 part of the gravitino -and then it loses protection and gets its yukawa coupling-, and another one where the piece left in the 44 is the set of three neutrinos, then getting its Majorana mass for the see -saw. Any of the two arrangements, if proved, should be a incredible success).
 The first approach at least gives us a very crisp hypothesis: That the almost massless fermions of the Standard Model are superpartners to the supergravity C-field (which Schwarz is calling A3 because it's a 3-form analogous to the 1-form A of electromagnetism). So I guess the homework exercise is to explain why that is or isn't a plausible statement... edit: OK, there are two versions of the homework exercise, one for "light fermions" and one for "charged fermions", as described on page 4 of your paper. second edit: Doing it for charged fermions (i.e. all of them except for the neutrinos) sounds a lot more natural. But then you don't get to explain the top-quark mass scale.

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Thanks for noticing the paper, I had not invoked it because it contains more general speculations, which could give support to the hypothesis but are independent of it.

 Quote by mitchell porter second edit: Doing it for charged fermions (i.e. all of them except for the neutrinos) sounds a lot more natural. But then you don't get to explain the top-quark mass scale.
Well, if we believe in the Higgs, it is not a scale but a coupling, which "just happens" to be equal to 0.998... If we think of it as a mass scale, I agree that it needs a second mechanism to explain why it is not at Planck's scale. Relying in orthodoxy, we could invoke Randall-Sundrum, can we?

If we are going first to try to attack the hypothesis, perhaps we could start with the typical questions on the relationships between fermions and gauge fields, and ask if the fermions (once we have broken down to D=4) can be in the same representation that the components coming from the C field. For usual susy, I think there are arguments about comparing the fundamental and adjoint representations, and perhaps they apply here too.

 Quote by arivero If we are going first to try to attack the hypothesis, perhaps we could start with the typical questions on the relationships between fermions and gauge fields, and ask if the fermions (once we have broken down to D=4) can be in the same representation that the components coming from the C field. For usual susy, I think there are arguments about comparing the fundamental and adjoint representations, and perhaps they apply here too.
Can you expand on this line of thought? Because it seems complementary to my own approach, which is to look for a known class of string models which looks promising (from the perspective of the hypothesis), and then to study what happens to the 11-dimensional fields in such models, and where the phenomenological fermions come from. This is the dumb strategy which says, OK, I don't know how to answer this question algebraically, but I can look for something similar that was already analysed, and learn from that.

This paper has (on page 3) a list of known ways to get something like the standard model from string theory. Now something which stands out for me, in our context, is option f, "M theory on G2-holonomy manifolds with singularities", because G2 has as a subgroup the 168-element group PSL(2,7), and there are a few papers on the arxiv by Stephen F. King proposing that this group is relevant for a "family symmetry" approach to standard-model masses, and 168 = 2 x 84. :-) And this paper, on the "G2 MSSM", says (page 7) "phenomenologically interesting G2 compactifications arise only for the case Q-P=3 and Peff=84", where those are all parameters determined by matter and gauge field representations.

Sorry for introducing more undigested numerology, when we're still just beginning to analyse the original idea, but I thought I should mention all that in case it connects with something you know.
 Blog Entries: 6 Recognitions: Gold Member I found the articles of Acharya, too. They seem very interesting. About the point of fundamental and adjoints, I had always though it basically as a "slot counting", ie if the fundamental is a 3 and the adjoint is a 8, obviously they do not match, stop. But I'd expect there is something deeper, looking at the Lagrangian. A thing that is beginning to puzzle me is the connection between a compactified theory and the non-compactified one. They do not have, generically, the same number of zero energy states, which is justified because some states can be extracted from the KK tower. But when does it happen such extraction, and how does it happen when you vary continously the radius of the compact dimensions, it is not explained in the literature.

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The following short explanation by Motl about the need of complex representations to explain chiral matter is relevant here:
 Quote by L.M. in motls.blogspot.com/2010/11/what-grand-unification-can-and-cannot.html However, there is a key condition here. The groups must admit complex representations - representations in which the generic elements of the group cannot be written as real matrices. Why? It's because the 2-component spinors of the Lorentz group are a complex representation, too. If we tensor-multiply it by a real representation of the Yang-Mills group, we would still obtain a complex representation but the number of its components would be doubled. Because of the real factor, such multiplets would always automatically include the left-handed and right-handed fermions with the same Yang-Mills charges! That's unacceptable because the left-handed fermions' properties differ from the right-handed ones. That's necessary for the parity violation in the weak interactions, the odd number 15 of the 2-component fermions we found, for CP-violation, and so forth. So the complex representations of the groups are totally necessary.
 Blog Entries: 6 Recognitions: Gold Member I liked yesterday presentation of Kaluza Klein theory by Lubos because of a specific point: he looks first at the classical field theory, so that instead of decomposing the graviton, you look how to decompose the metric. Explicitly, he points out that the 5D metric decomposes in a 4D metric plus a 4D vector plus an scalar. As a D-dimensional metric has D(D+1)/2 components, and a 4D classical vector has four components, you see that the decomposition matches as expected. Now, think about this: how many vectors can you get, at most, out of a D-dimensional metric when producing a 4 dimensional one. Not a lot. In fact only for 5 D you can get the maximal isometry group, U(1). Already in 6D, a two-sphere has SO(3) symmetry, thus three generators and you need 12 components for three 4D vectors. But the 6D metric has 21 components only, and you need 10 of them for the 4D metric. Because of it, all the Kaluza Klein "sphere compactifications" beyond electromagnetism need a closer examination. (btw, it is interesting that in this classical view, the number of components obtained from 6D to 4D is the same that going from 11D to 10D. And it is also peculiar that in 9D you have 45 components, so 35 free, so a maximum of 8 vectors "and a bit" again, while you need 9 vectors to implement SU(3)xU(1), the symmetry of the 5-dimensional manifold CP2xS1. I 'd like to see how all of this translates to the quantum view, with the graviton instead of the metric)
 The realistic string models are quite complicated, with CYs, "fluxes", singularities, etc. The standard model fields often live on one brane among many (e.g. susy might be dynamically broken on another brane, and the effect is then transmitted to the observable "GUT brane" by fields in the bulk). The simpler scenarios, like compactification on a sphere, would have been ruled out long ago by arguments like the one you just made. They are still theoretically and pedagogically interesting, but too simple for reality. Something I'd like to have is a geometric interpretation of the gravitino field, and (related) an understanding of how the M-branes interact with the gravitino field. I think there is a bilinear interaction between fermions on the brane and fermions in the bulk, but haven't found a good reference.

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 Quote by mitchell porter In particular, reference 7 is full of "84+84"s and "28+56"s. But it might take a while to assimilate.
Definitely is not so "homework". I have been perusing some huge papers on compactification of D=11 N=1 SUGRA down to D=4, N=4. The antisymmetric A_{pqr} is divided in different portions, and particularly the 4 dim "pseudovector", where the first component lives in 4 dimensions and the second and third components take indexes in 7 dimensions, so 7*6/2=21 different "pseudovectors". The usual technique is to join them with another seven (where only the last index lives in seven dimensions) to build one of these 28, and then go to look for "hidden local gauge symmetries" based in SU(8) and E7. Pretty stuff.

So really there is a symmetry in the A_{pqr} surviving in 4D. But they do it in a obscure way, without connecting it to the fermionic states, and besides in a compactification manifold very trivial, which fails to produce the standard model gauge group.
 Dear Mitchell, I don't want to disappoint you too much but the "research project" you're proposing is a classical textbook example of numerology which is not really rationally justified. Numerology is based on focusing on a single number - for example, the number of particle species in your case - and neglecting all other properties of the objects that the number is related to. In particular, there are 84 physical components of the C-field in M-theory because they transform as the antisymmetric 3-form in 11D. Counting transverse oscillations only, that produces 9 x 8 x 7 / 3 x 2 x 1 = 84 components. But it's not just the number 84 that is important; it's the whole 84-dimensional representation under SO(9), the little group, that matters. Clearly, there is no half-spin (to become fermionic) 84-dimensional irreducible representation of spin(9). If you had another set of 84 states, they would have nothing to do with the 84 states of C-field in M-theory. There exist no phase transitions in M-theory that could "change everything" about physics while preserving the number of massless excitations. In particular, you wouldn't get the correct Standard Model numbers by any reinterpretation of the 84 states. To get a Standard Model from M-theory, you need more complex treatment - compactification with everything it needs - and such a treatment changes both the gauge groups and the quantum numbers of the states - as well as the number of these states. It just makes no sense to be excited by the appearance of a number such as 84 at two places, especially if one of them is really adding apples and oranges (leptons and quarks) to get this result. The probability that a random number below 100 is 84 is 1% - which is not even enough to claim a 3-sigma evidence that "something is going on". One needs many more things to agree. Best wishes Lubos
 Blog Entries: 6 Recognitions: Gold Member :-DDD Lubos I am surprised that you have considered to waste ten minutes to write a comment in this thread. I hope that you agree at least that 84+44 lives in the same supermultiplet that the 11-D fermion. The questions are, from less to more hypothetical: - is there any special symmetry to characterize the set of states one gets by applying the supersymmetry transformation to the bosons in the 84 multiplet? - given such symmetry, is it possible to devise a mechanism to protect the mass of these states? - given the symmetry and the mechanism, does it survive down to 4 dimensions? - given the 4 dimensional protected fermions, are they related to the 84 light fermions of the standard model? In fact the last question is going to fail because in 11D Kaluza Klein the groups do not include B-L. But all the other steps are interesting from the point of view of "exercise", as Mitchell says. Regrettably, neither him nor me have the required acquittance and skill with the tools.

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 Quote by lumidek It just makes no sense to be excited by the appearance of a number such as 84 at two places, especially if one of them is really adding apples and oranges (leptons and quarks) to get this result.
In any case, M-theory or not, we have the question of the naturalness of Yukawa couplings. Of the 12 different values coupling the 96 fermion states, we have 11 "almost zero" values for the couplings between 84 different states, and they should have somewhere a symmetry protecting them. It could be a 21 or a 42 of something, but it should be there. I am also interesting on hearing of realistic candidates for this group, or other ways to solve this naturalness problem.

EDIT: btw, there is also a trivial way for the 84 to appear in two places... it is also the number of components of the $$A_{pqrstu}$$ tensor Branes and pentabranes.

Hi lumidek and arivero. Alejandro, this would be our mistake:
 Quote by arivero the set of states one gets by applying the supersymmetry transformation to the bosons in the 84 multiplet
But there's no such thing as "the supersymmetry transformation". The superalgebra has more than one fermionic generator.
 Blog Entries: 6 Recognitions: Gold Member Hmm why? In 11D it is N=1. What happens is that down to 4D it can generate up to N=8, depending of the compactification manifold.

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