M-theory phenomenology - homework exercise

[mitchell porter]

In the current long thread discussing string theory, https://www.physicsforums.com/showthread.php?p=2982921":

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

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?

 

[arivero]

I do not hope to get too much from the exersice, but let's 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 (http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+EA+%22BOYA%2C+L%22+OR+EA+%22BOYA%2C+L+J%22+OR+EA+%22BOYA%2C+LOUIS+J%22+OR+EA+%22BOYA%2C+LUIS+J%22&FORMAT=www&SEQUENCE=ds%28d%29 ). 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.

 

[mitchell porter]

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, http://arxiv.org/abs/hep-th/0611314" is full of "84+84"s and "28+56"s. But it might take a while to assimilate.

 

[arivero]

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)

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).

 

[mitchell porter]

The first approach at least gives us a very crisp hypothesis: That the almost massless fermions of the Standard Model are superpartners to the http://ncatlab.org/nlab/show/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 http://arxiv.org/abs/0910.4793" .

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.

 

[arivero]

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.

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.

 

[mitchell porter]

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.

http://arxiv.org/abs/hep-th/0406228" , on the "G2 MSSM", says (page 7) "phenomenologically interesting G2 compactifications arise only for the case Q-P=3 and P_eff=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.

 

[arivero]

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.

 

[arivero]

The following short explanation by Motl about the need of complex representations to explain chiral matter is relevant here:

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.

 

[arivero]

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)

 

[mitchell porter]

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.

 

[arivero]

In particular, http://arxiv.org/abs/hep-th/0611314" 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.

 

[lumidek]

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

 

[arivero]

:-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.

 

[arivero]

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.

 

[mitchell porter]

Hi lumidek and arivero. Alejandro, this would be our mistake:

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.

 

[arivero]

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.

 

[mitchell porter]

Yes, but even an N=1 theory has a whole set of supercharges Q_a with a spinor index. (For N>1 there's a further index r, as in Q_ar, r=1...N.) In eleven dimensions there are 32 of them! (Though this number reduces in practice.) So when you want to map the 84-dimensional space of bosonic states arising from the 3-form, into the 128-dimensional space of fermionic states of the gravitino, which supercharge do you use?

edit: There's got to be some simple algebraic/geometric way of grasping how these 2⁽²⁺³⁾ supercharges act on (2-form + 3-form), but I don't have it yet. The discussion in Weinberg volume III of how the states are built up looks promising.

 

[arivero]

It seems that the survival of degrees of freedom down to 4D depends of the Betti Numbers of the compact manifold.
Looking at arxiv 10103173, it seems that the appropiate sequence should be (b0,b1,b2,b3)=(1,0,14,49).

Has anyone got a reference to the betti numbers of S3xCP2 and similar manifolds proposed by Witten 1981?

 

[mitchell porter]

I can't access http://iopscience.iop.org/0264-9381/1/5/003" right now, but it seems to say b2=1 for all of them.

Some of the G2 manifolds in Joyce's book (chapter 12) are close to what you want. But for both the Joyce manifolds and the Witten manifolds, everything would be nonchiral. And the counting is very different for Acharya's singular G2 manifolds e.g. the gauge symmetry is enhanced by extra states trapped on the singular surfaces of the compactification space.

Anyway, even when I use the formulas in Duff and Ferrara, I don't understand why you chose those values.

 

[arivero]

I will try to access from the campus, we have the printed version, not sure about the electronic one, of this journal.

Look at tables IX and V of duff ferrara. With those values you get pretty close to the standard model content. Without the higgs, the SM is 48 "WZs", and 12 "vectors". Setting b1=1, b2=0 and asking the two last equations in table V to keep the values of the torus for f and rho, you get this result. (EDIT: in any case, I am not sure about Duff Ferrara, because when I put in the Betti numbers of a 7-sphere, naively, it fails to produce the expected number of symmetries).

Still, you are right that it is a bit against our try, because these vectors do not come really from the graviton, if we believe table V fully.

EDIT: Great, on-campus access works! Amazing. For some journals, specially the most veteran private printers (Elsevier and Springer), printed access does not imply online access. Still, I think that there was an exception to the rule, CP2xS3

 

[arivero]

Well, I found the relevant parragraph. It is not exactly as I hoped, but still it does not coincide with Duff Ferrara nor with D'Auria Fre

"Mpqr... for most of these spaces, the first and third Betti number vanish, so one does not get massless scalars [from the Aijk tensor]. (The exceptions are CP2xS3 and CP2xS3/Zk, for which the third Betti number is one and one gets one massless scalar in four dimensions; and CP2xS2xS1, for which the first and third Betti numbers are one and one gets two massless scalars.) However, for the Mpqr the second Betti number is one (except for CP2xS2xS1, where it is two), so one would get one (or two) massless spin one particles (in adition to the gauge fields of SU(3)xSU(2)xU(1) coming from the metric tensor)"

It adds that "these massless spin one particles do not have minimal couplings to any matter fields; They interact throught derivative couplings, such as magnetic..."
And that "there is, however, an old theorem that zero modes of antisymmetric tensor fields are always neutral under any continuous symmetries."

 

[arivero]

Could someone check the parragraph I copied from Witten's shelter island talk? Specifically, it seems to me that the third Betti number of CP2xS2xS1 is 2, not 1:

- The betti numbers of CP2 are 1 0 1 0 1
- The betti numbers of S2 are 1 0 1
- The betti numbers of S1 are 1 1
So the product of CP2 times S2 should have 1 0 2 0 2 0 1, and the
betti numbers of CP2xS2xS1 should be 1 1 2 2 2 2 1 1. The third betti number is the fourth in the list (as it starts with b0) and it is 2.

 

[arivero]

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.

Hmm, an observation here. If instead of the higgs, you put directly massive W and Z, you get exactly 128+128. This is because a massive vector supermultiplet includes, besides the extra degree of freedom of the vector, a new scalar, and then a chiral fermion to act as a superpartner of them.

So now you have the following "slots":

96 squarks and sleptons and 96 spin 1/2 quarks and leptons
6 spin 1 and...
2 spin 0 W, plus 8 spin 1/2 partners
3 spin 1 and...
1 spin 0 Z, plus 4 spin 1/2 partners
2 graviton and 2 spin 3/2 gravitino.
2 photon and 2 photino
16 gluons and 16 for the gluinos

total 128 + 128.

 

[mitchell porter]

Such a model is described in a 1984 paper by Pierre Fayet, "dx.doi.org/10.1016/0370-1573(84)90113-3"[/URL] (see table 2).

edit: Wait, is that just the MSSM again, but with a change of variables?

 

[arivero]

edit: Wait, is that just the MSSM again, but with a change of variables?

It is MSSM but Fayet notation helps to stress that the infamous five higgs bosons of the MSSM do actually come in two packages: three of them (w+,w-,z) from the "effective theory" of massive gauge bosons, and two of them (h0 and h'0) as a consequence of the higgs mechanism itself.

It is a bit obscure to relate Fayet notation with the standard (eg from http://pdg.lbl.gov/2010/reviews/rpp2010-rev-susy-1-theory.pdf ) but it is the same, yes. That was the point of Boya preprint years ago, to notice that the MSSM has 128+128. What is new to me is that now we see an obvious candidate (the pair h0,h'0) to be removed from the set in order to make place for the graviton instead.

Edit: from reading the PDG review (and also an exchange with LM), it seems that people prefers to consider the massive supermultiplet as a pair of massless supermultiplets where the spin 1/2 particles can mix and then the mass eigenstates can be different. I am not sure if this amounts to a further breaking of supersymmetry. And, it does not seem right to give different mass to the charged wino and the charged higgsino, neither to claim that you are mixing them: they already live in the same massive supermultiplet! perhaps the so-called mixing is really a trick to bypass the need of having higgsinos in the fundamental and the winos in the adjoint representation of the gauge group.

 

[mitchell porter]

The return of the M2-brane:

At Physics Stack Exchange, http://physics.stackexchange.com/questions/6303/is-our-universe-a-21d-superconformal-field-theory" to use the "AdS4/CFT3" version of gauge/gravity duality for phenomenological purposes. On the gravitational (AdS) side, such theories involve a stack of M2-branes.

http://arxiv.org/abs/0909.2036" is a rather technical paper about CFT duals to some AdS4 solutions of M-theory - but note the reference to "G-flux". That's the field strength of the C-field 3-form we were talking about in this thread! So take heart Alejandro, you might yet be able to do some "M2-brane phenomenology". :-)

edit: Or just see http://arxiv.org/abs/0807.4924" introducing the ABJ model (variation on ABJM) - C-field shows up, starting on page 7.

 

[mitchell porter]

Alejandro just brought up this observation of Boya's (128+128 states in MSSM and in N=1 d=11 sugra) at Matt Strassler's blog, and asked if we could somehow replace the MSSM higgs with a graviton in order to get a MSSM-like theory from a proper supergravity multiplet. From a contemporary perspective this line of thought seems unmotivated, since string theory allows you to arrive at a low-energy theory in completely different ways; the numbers 128 and 256 aren't so special any more. Still, I will record an idea: I think the best way to "replace the higgs" would be (if it is possible) to use a scalar gravitino condensate. What's appealing about this, is that it would relate the fermion masses to quantum gravity, and we would still have a scalar that could be the 125 GeV object seen at the LHC.

 

[mitchell porter]

One more remark. Suppose we compare the N=8 supermultiplet (which you get from the maximally symmetric compactification of d=11 sugra) to the MSSM particle spectrum, by counting the number of states with spin 2, 3/2, 1, 1/2, 0. For N=8 you get

2, 16, 56, 112, 70

while for the MSSM you get

0, 0, 24, 96+24+8, 96+8

where 24 counts gauge bosons and gauginos, 96 counts SM fermions and their sfermion superpartners, and 8 counts higgs and higgsino states.

This gives an idea of the redistribution of states that would be necessary to get a modified SSM from the N=8 multiplet, or alternatively, to get a "truncated MSSM plus gravity" from some other compactification of the d=11 theory. As Strassler points out, there would still be further problems in making this idea work - how do you get chiral fermions? If one somehow got to that stage, I think the next concept to investigate would be torsion.

But continuing at the kindergarten level of moving numbers around, to make an "N=8 SSM" from the MSSM, the MSSM needs to gain a graviton, 8 gravitinos, and 16 new gauge bosons, and it needs to lose 16 spin-1/2 degrees of freedom and 34 spin-0 degrees of freedom.

Conversely, to make an "N=1 SSM with gravity", you just want one graviton and one gravitino (i.e. mSUGRA), the equivalent of losing one chiral superfield. Maybe there are only two right-handed neutrinos, rather than three. But what does that do to anomaly cancellation? Nothing I guess, since it's a singlet under the SM gauge group...

The N=2 SSM has too many states for this exercise - N=1 SSM, plus all the mirror particles. But what's the counting like if you just have N=2 SQCD plus gravity, and count on leptons and electroweak to emerge via Seiberg-like duality (one version of the sBootstrap idea)? The counting is (for one graviton; two gravitinos; one graviphoton and eight gluons; six flavors of quark and six flavors of mirror quark and eight Dirac gluinos; squarks, mirror squarks, and sgluons)

2, 4, 2+16, 48+32, 48+16

Well, there's plenty of room, e.g. to add a U(1) for the sBootstrap charges and/or maybe even to add a hidden susy-breaking sector. And I note that we now have the magic number of 84 states, as at the start of the thread! Though if they are to be identified with the "C-field" and its superpartner, it would make more sense for the "2,4,2" to come from the additional U(1) supermultiplet, rather than from the N=2 graviton multiplet.

When I made this "M-theory homework thread", 18 months ago, I wanted to treat this particular idea of Alejandro's as an exercise, such as might be posed to a graduate student, in the hope that we would actually prove, one way or another, whether it could work. Here we finally have the beginning of a systematic enumeration of cases. Maybe we'll even get to the proof, given another 18 months...