# N=8 cosmology

## Main Question or Discussion Point

My ongoing study of https://www.physicsforums.com/showthread.php?t=483871" that this has something to do with the fact that the shear viscosity to entropy ratio of a strongly coupled N=4 super-Yang-Mills plasma is 1/4π.

Sheppeard has her own program of deriving field theory from categorical quantum gravity, but I'm not going to talk about that here. Instead, I want to speculate on how one might try to get Riofrio's numbers (perhaps just as a first-order approximation) from a known field theory.

"3/4π" is the central number here. That's the dark matter fraction, the dark energy fraction is exactly three times that, and that already accounts for 96% of the universe. So it could be that we have four copies of the same physics, each of them accounting for 3/4π of the total energy density; but one of them is in "dark matter" mode, and the other three are in "dark energy" mode.

In my attempts to find a field-theoretic analogue of Sheppeard's particle physics, I was driven to consider N=2 gauge theories. In such a theory, there will need to be a lot of heavy particles - for each visible particle species, there will also be a superpartner, a mirror partner, and a mirror superpartner, and all of these (on a conventional understanding) will need to be heavy. Cosmologically, one would therefore expect the dark matter to consist of the lightest stable super- or mirror particles. In other words, the dark matter sector consists of stable heavy particles from a N=2 gauge theory.

But in order to interpret Riofrio's numbers, I just speculated that the dark energy sector consists of three almost-copies of the dark matter sector. Thus, in a very simple way, I am led to wonder whether Riofrio's numbers can somehow come from d=4 N=8 supergravity.

Before superstrings, there was a lot of interest in N=8 supergravity as a theory of nature. The spectrum doesn't look much like the real world, so one has to treat it as a preon theory. I believe that one reason it fell out of favor was the apparent impossibility of getting chiral fermions, the consideration which also sank Kaluza-Klein models based on d=11 N=1 supergravity. Also, it was believed to be divergent, and therefore not fundamental.

In recent years, new methods of calculation have suggested that N=8 supergravity is perturbatively finite after all, and it has even been proposed that it is http://arxiv.org/abs/0808.1446" [Broken]]

• identifies SU(3) with the diagonal subgroup [SU(3)color × SU(3)family]diag
• shifts all U(1)em charges by a spurion charge 1/6.

Of course, for this to work there would have to be new and very strange dynamics, where the weak interactions would have to be dynamically generated with composite W and Z bosons, while the gluons and the photon would be elementary. Furthermore, the family symmetry SU(3)family does not commute with weak isospin SU(2)w in the above scheme. Looking at the diagonal subgroup may likewise appear a strange thing to do, but according to http://arxiv.org/abs/hep-ph/0003183" [Broken] such ‘flavor color locking’ may actually occur, and not only in strongly coupled QCD! So it is not completely excluded that an ‘obviously wrong’ theory (or some extension of it) could turn out to be right after all – just like QCD would have been considered ‘obviously wrong’ had it been proposed in 1950 as a theory of strong interactions, and without knowledge of its underlying dynamics![/QUOTE]
This sounds a lot like what we have been looking for in the search for a theory that produces https://www.physicsforums.com/showthread.php?t=485247", where I have been considering the possibility of a perturbed version of the self-duality of N=2 Nf=6 SU(3)c gauge theory. And this "stationary point with residual SU(3) × U(1) symmetry" even has N=2 supersymmetry!

So what is the cosmology of N=8 supergravity? I read somewhere that it develops a large negative cosmological constant without finetuning. But couldn't we apply Weinberg's anthropic argument for a small positive cosmological constant? As for the dark matter, perhaps some version of http://arxiv.org/abs/hep-ph/0202161" [Broken], solitonic diquark condensates containing a trapped exotic phase of QCD, can do the job. It seems somehow concordant with Nicolai's own appeal to exotic dynamics.

This may just be the first step on a long road; "the simplest quantum field theory" seems to be very hard to study. But the reward could be very great.

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Related Beyond the Standard Model News on Phys.org
It turns out that the approach to N=8 sugra phenomenology mentioned by Nicolai originates with a talk given by Murray Gell-Mann in 1983. You can see the talk on http://tuvalu.santafe.edu/~mgm/Site/Publications.html" [Broken] - item 94, "From Renormalizability to Calculability" - but be careful, that's a 24 Mb scanned PDF. It's the "last-ditch effort" to avoid composite fermions in N=8 sugra phenomenology, mentioned towards the end of the talk (page 9 of the PDF, upper right).

Much more recently, http://arxiv.org/abs/0905.3943" [Broken].

I will try to be cautious, rather than proclaim that this is The Answer (and that Gell-Mann knew it back in 1983). But it takes everything I've been talking about to a higher level - as well as conveniently providing a distinguished pedigree.

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arivero
Gold Member
It turns out that the approach to N=8 sugra phenomenology mentioned by Nicolai originates with a talk given by Murray Gell-Mann in 1983. You can see the talk on http://tuvalu.santafe.edu/~mgm/Site/Publications.html" [Broken] - item 94, "From Renormalizability to Calculability" - but be careful, that's a 24 Mb scanned PDF. It's the "last-ditch effort" to avoid composite fermions in N=8 sugra phenomenology, mentioned towards the end of the talk (page 9 of the PDF, upper right).
Good finding! I searched for this paper, as I was sure I had read about Gell-Mann SU(3) diagonal somewhere, but it was not available in KEK, and Spires was not very informative. It seems that there is a related talk in Erice 1983 lectures, "Can Supersymmetry Be Connected To Observation?"

It was really a last ditch effort, as it seems from the publication record that Gell-Mann left for the Santa Fe institute soon after this work, not pursuing particle theory anymore.

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I'm now studying the details of Gell-Mann's idea (the details are in Nicolai and Warner, reference 25 above).

Brannen and Sheppeard have been working towards a fundamental quantum theory by starting with the "data" of the particle masses and mixing matrices, expressed in terms of circulants and their eigenvalues (an approach inspired by the Koide relation), and working "backwards" towards a formalism, with various algebraic ideas (twistor gauge theory, Schwinger's measurement algebra) as their guide and inspiration. Gell-Mann, on the other hand, started with a field theory (N=8 supergravity) and looked for a way to embed the gauge groups and representations of the standard model.

If all of these ingredients did mesh successfully, that would really be a theory of everything. What I expect to discover is that they don't mesh; but we'll be able to say why they don't. And knowing the reasons why, should clarify many things.

I am going to devise some barbarous terminology, in order to point out a curious feature of Gell-Mann's construction. If some of the words I employ sound a little kooky, remember that they're just names for the mathematical objects.

N=8 supergravity contains one big SO(8) supermultiplet with one spin-2 state (graviton), eight spin-3/2 states (gravitinos), 28 spin-1 states (gauge bosons), 56 spin-1/2 states ("spinors", "ordinary" fermions), and 70 spin-0 states (scalars). Gell-Mann is concerned with a vacuum in which SO(8) is broken to SU(3) x U(1). We can then classify all these states from the N=8 supermultiplet in terms of their transformation properties under this SU(3). They will turn out to transform under 1-, 3-, 6- and 8-dimensional N=2 representations: singlets, (anti)triplets, (anti)sextets, and an octet. The details are in equation 2.10 of Nicolai and Warner.

I should point out that this SU(3) is apparently not color, but has something to do with color. In the quote above, Nicolai proposes that it is the diagonal subgroup of the product of color SU(3) and "family SU(3)", which would act on the generations. But it might be safer just to call it a "mystery SU(3)" that has something to do with both color and family.

In terms of mystery-SU(3) representations, the up-type quarks are 8+1, the down-type quarks are 6+3bar (or 6bar+3), and the leptons (charged and neutral) are all just 3 or 3bar. These singlet, triplet, sextet, octet... representations are part of larger N=2 supermultiplets which extend to other values of spin.

So here comes the barbarous terminology. I will refer to the singlet and octet supermultiplets as the "electrostrong sector", and the triplet and sextet supermultiplets as the "leptodown sector". This allows us to say certain things. For example, we can say that the up-type quarks are "electrostrong fermions". We can classify the 70 scalars into electostrong scalars and leptodown scalars. And we can also observe that there are massive leptodown gauge bosons, and massless electrostrong gauge bosons.

Now a word about the graviton and the gravitinos. In terms of mystery-SU(3), there are two singlet gravitinos, and a triplet and an antitriplet (accounting for all eight gravitinos). The singlet gravitinos form a separate N=2 multiplet along with the graviton and one of the gauge bosons (I suppose that gauge boson is a "graviphoton"). The (anti)triplet gravitinos are part of the "leptodown sector": the sextet multiplets only contains spinors and scalars, but the triplet multiplets extend all the way into spin-1 and spin-3/2. (I should add that this analysis all assumes anti-de-Sitter space - something that will have to be amended later - and these are a type of N=2 multiplet peculiar to AdS.)

Among the spin-0 scalars, there are goldstones that will give mass to some of the N=8 gauge bosons, and among the spin-1/2 scalars, there are goldstinos that will give mass to all the gravitinos. Using the barbarous terminology, the goldstones are all leptodown scalars (in triplet and antitriplet representations), responsible for making the leptodown gauge bosons massive.

As for the goldstinos, there is a triplet, an antitriplet, and two singlets, accounting for all the gravitinos. It is part of the elegance of this scheme that all the remaining N=8 spinors correspond exactly to the elementary fermions of the standard model (assuming three right-handed neutrinos). Nothing else is left over.

Sheppeard, working with her categorified version of the Bilson-Thompson braid correspondence, has posited the existence of additional "mirror neutrinos". In her scheme, which she sometimes calls "neutrino gravity", neutrinos are created at cosmological horizons and are responsible for the introduction of mass (she has no Higgs), via the electroweak sector. You can read about it http://vixra.org/abs/1010.0029" [Broken]. Also, she rejects standard notions of dark matter and dark energy; they are to be understood as holographic quantum-gravity effects, and therefore as another aspect of "neutrino gravity".

Here is the curious thing. In Gell-Mann's proposal for N=8 supergravity, the neutrinos (like the charged leptons) transform under 3 and 3bar. The extra spin-1/2 fermions, the goldstinos that give mass to the gravitinos, consist of two singlets, a 3, and a 3bar (matching the transformation properties of the gravitinos). So it looks as if Gell-Mann's goldstinos are Sheppeard's mirror neutrinos - a 3+1 and a 3bar+1 - and that the singlet mirror neutrinos give rise to "dark matter", and the (anti)triplet mirror neutrinos give rise to "dark energy"! (Recall the starting point of this thread: there is three times as much dark energy as there is dark matter.)

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I posted https://www.physicsforums.com/showthread.php?t=539315" about "Cecotti-Cordova-Vafa" theories - 3-dimensional N=2 gauge theories, which arise on a domain wall of a 4-dimensional N=2 theory, and which are associated with a braid describing the spacelike "R-flow" of the Seiberg-Witten differential across the domain wall. I wrote that this was obviously of interest in trying to make Bilson-Thompson's correspondence work, but that one faced the immediate problem of needing a 4-dimensional theory for phenomenology, not a 3-dimensional theory.

In http://arxiv.org/abs/0905.3415" [Broken] for more on this.)

In an earlier comment, I mentioned that Changhyun Ahn had studied the RG flow for N=8 sugra near this Gell-Mann-Nicolai-Warner fixed point, in terms of an M2-brane worldvolume theory governed by three mass parameters. It doesn't take much intuition to imagine that all of this may provide a way to interpret the N=8 Gell-Mann theory in terms of a Bilson-Thompson correspondence. All we need is an M-theory embedding of N=8 sugra, in which an Aganagic uplift of Ahn's RG-flow provides a timelike realization of Cecotti-Cordova-Vafa R-flow braiding of the masses (VEVs of N=8 scalars, presumably). :-)

Aganagic wrote a paper about Khovanov knot homology recently, so she might be just the person to elucidate this issue.

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1) I have hoped for a while to explain Alejandro's sBootstrap in terms of a Seiberg duality, in which all the leptons arise as elements of Seiberg's emergent meson superfield. http://arxiv.org/abs/hep-th/0505153" [Broken] should explain what I mean. See the very first section, in which Strassler sums up Seiberg duality as an equivalence between "SQCD" and "SQCD + M". The "M" is the meson superfield which arises in the dual description, and that's where the leptons and the weak interactions arise, according to this idea.

Now visit page 33 of Strassler's article. He mentions the N=2 SU(N) gauge theory with Nf=2N flavors, which is self-dual; and that you can get the N=1 theory by adding a mass term. Obviously, the case where there are three colors and six flavors starts to look like the real world; except that we want an "N=0" version of the duality. There are steps towards this in the recent literature, especially as a result of Zohar Komargodski's "interpretation of Seiberg duality" in terms of phenomena from QCD.

Since we have these two SU(3)s appearing in Gell-Mann's construction, color SU(3) and family SU(3), I have to wonder whether there's a Seiberg duality buried in there somewhere, which might even explain the remaining mysteries of the construction, such as where the weak interactions come from. Perhaps there's an alternative way to assign representations and hypercharges to the N=8 fields which corresponds to the other side of the duality, and which is related to the version we're exploring here by an electric-magnetic duality of N=8 supergravity.

In this regard, it might be of interest to look at the twistorial formulation of N=8 supergravity (see the paper I linked earlier, on "the simplest quantum field theory"; also see Andrew Hodges's recent work on twistorial N=7 supergravity). The Nsusy=2, Nc=3, Nf=6 can be derived from a http://arxiv.org/abs/0708.1248" [Broken]...

2) In a sense, this whole search derives from Koide's mysterious formula for the charged lepton masses, which Brannen and Sheppeard then generalized to all the fermions. Koide has written a few papers (one just a few days ago) in which he tries to get his relation from a U(3) x O(3) family symmetry. N=8 supergravity actually has an SU(8) x SO(8) symmetry, and so I'm wondering if Koide's family symmetry descends from that; perhaps it governs Ahn's RG flows.

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arivero
Gold Member
SU(8) x SO(8) is intriguing, I have never understood it and why the SU(8) is hidden. I think I have a personal problem with global vs local gauge theories, perhaps a problem of translation Spain/USA. And while the hidden SU(8) was a popular topic in the literature time ago, I have never heard of it from any teacher.

Still, it could be a better resource than SU(3) from SO(8). A problem I had about the later is that I see SO(8) as the isometries of S7, and this view limits my enjoyment to some subgroups. For instance SO(5)xSO(4) is pretty natural, because I can see how the sphere S7 is composer of S4 times S3. But for SU(3), which are the isometries of CP2, I need to see a way to fiber over CP2 and get S7, or to compactify three dimensions of S7 and get CP2.

I believe it's just that the CP3 of S7 gets squashed, so that isometry is preserved only for a CP2 within the CP3. See hep-th/0208137, section 5.2. Just like, if you squash S2, all that's left is the U(1) of rotation along the equatorial S1. Also, S1 x squashed CP3 is equivalent to "an S3-bundle over S4 with gauge potential" (arXiv:0809.3684, section 2); that must mean something. (btw, I don't know what the cause of the squashing is supposed to be - heavy objects at the opposite poles of the S7?)

Congratulations on being dubbed the new Eddington by Lubos. Along those lines, I have an alarming thought about N=8 and the sBootstrap. If you count states for N=8 supergravity, there are 256 of them. But if you count representations, there are 163, including 99 bosons (1 graviton, 28 gauge bosons, 70 scalars). Now, SU(10) has 99 generators, and it is the Pauli-Gürsey symmetry for five flavors, which is the closest thing we have to a derivation of the sBootstrap. That is, for QCD with 2 colors and 5 flavors, chiral symmetry breaking leaves a residual 55-dimensional symmetry, Sp(10), and then there are 44 Goldstone bosons, 24 of them pions, 20 of them "2-color baryons". This is described on page 6 of hep-ph/0501200; the idea of that paper is that this exact symmetry of 2-color QCD should still be an approximate symmetry of 3-color QCD, and the "2-color baryons" simply become diquarks. The resulting inventory of pions and "goldstone diquarks" is very close to the combinatorics of the sBootstrap. If the sBootstrap can be derived from a Seiberg duality (S is for Seiberg!), one might expect it to involve quasi Goldstone fermions that are superpartners of these 44 objects.

All of that is so logical (relatively speaking) that I hesitate to confuse the issue by then turning back to the "99 bosons" of N=8 supergravity. But I'll do it. After all, in this thread I'm speculating that the sBootstrap can be realized at the Gell-Mann-Nicolai-Warner critical point. Unfortunately, I cannot see any sane way to identify, or even just canonically associate, the 99 "N=8 bosons" with the 99 generators of Pauli-Gürsey SU(10).

arivero
Gold Member
Missed the reference (ah, no, ok, I saw it) of Lubos, but time ago there was some comparisions to Ridberg, claiming that Ridberg was of course a crackpot. The main argument against the things I do is to paraphase Feynmann in the "lockbox number argument", but it is not that I am a passing-by, I have been in the room seeing him (Feynmann, no Lubos) to try to open the box for yars. For decades.

Anyway, as Eddington (btw, did I calculated the fine stricture constant? Hey, was not me , it was Hans, I only did the thread here). could say: 128=44+84. So 44 is most probably just the difference 128-84. The key numbers are the total number of helicities in these 99 massless bosons, 128, and in the M2-brane or M5-brane source field, with 84.

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arivero
Gold Member
SU(10) has 99 generators? Hmm I was looking to SO(10), and I failed to notice this one, of course, 5^2-1.

I had suspected of some quotienting going on, particulatly from these 28 gauge bosons, as SU(3)xSU(2)xSU(2) has 14. Yep now it seems that we are doing numerology at its best, in the sense of Ramond 1974 (to quote the oldest reference I know to this use of the word).

But now we can check the numerology against a model - Gell-Mann's assignment of the SM fermions, to states existing at the SU(3) x U(1) fixed point of N=8 supergravity. That's really the raison d'etre of this thread - to see whether a few physical postulates (the dark sector from N=8 gravitinos, Koide mass triplets from Ahn's RG flow, a Seiberg duality between SU(3)color and SU(3)family) could possibly be true of this model.
Anyway, as Eddington (btw, did I calculated the fine stricture constant? Hey, was not me , it was Hans, I only did the thread here). could say: 128=44+84. So 44 is most probably just the difference 128-84. The key numbers are the total number of helicities in these 99 massless bosons, 128, and in the M2-brane or M5-brane source field, with 84.
https://www.physicsforums.com/showthread.php?t=447612" was that all the SM fermions, except for the top quark, are d=11 superpartners of the C-field. At the time I pointed out that there isn't a unique superpartner. But your idea has much more bite if considered in the context of Gell-Mann's model, which (as elaborated by Nicolai and Warner) associates each SM fermion with part of a specific N=2 multiplet. We can then look at how they embed into the full N=8 superalgebra, and at how they look from 11 dimensions, and see if there really is a supersymmetry generator which maps the top to the graviton, and all the other SM fermions to the C-field.

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A brief update... My current thinking is that dark energy is (in some sense) the holographic shadow of the hadrons, and dark matter is the holographic shadow of the leptons. By a holographic shadow, I mean a manifestation in four dimensions of the higher-dimensional couplings of the particles. All the gravitinos in four dimensions descend from the single 11-dimensional gravitino. So I'm supposing some sort of universality in the interactions between the gravitinos and the standard model fermions. The fact that there are three times as many quarks as there are leptons (counting quarks with the same flavor, but red, blue or green color, as three types of quark) is correlated with the division of Gell-Mann's gravitinos into six SU(3) (anti)triplet gravitinos and two SU(3) singlet gravitinos, and with the fact that dark energy contributes about three times as much to the energy density of the universe as does the dark matter. Possibly the "dark energy gravitinos" form a condensate which couples to the boundary of "hadron bags", while the "dark matter gravitinos" are decoupled from this condensate and have a different equation of state.

edit: If we are to think in terms of extra dimensions, it makes more sense the other way around: the hadrons are the "shadow" of the triplet gravitino condensate, the leptons are the "shadow" of the singlet gravitinos. The 11-dimensional gravitino constitutes the entire multiplet of fermionic states in 11-dimensional supergravity, so this amounts to saying (1) Gell-Mann's quarks and leptons have canonical ties to the triplet and singlet d=4 gravitinos, respectively (2) dynamics in the higher-dimensional bulk dominates: the observable interactions among the standard-model fermions are just froth on the bulk interactions among the gravitinos.

Here is how the N=8 supergravity multiplet breaks up at the point with SU(3) x U(1) symmetry:
Nicolai said:
Since the unbroken group symmetry at the stationary point is SU(3)xU(1), the fields of the N = 8 theory, originally transforming in SO(8) representations, must now be decomposed into SU(3)xU(1) representations. Following [Gell-Mann], we assign the hypercharge y = ½ to the gravitino [...] We get
$s=2: 1 → 1(0)$
$s=\tfrac{3}{2}: 8 → 1(½) \oplus 1(-½) \oplus 3(\tfrac{1}{6}) \oplus \bar{3}(-\tfrac{1}{6})$
$s=1: 28 → 1(0) \oplus 1(0) \oplus 8(0) \oplus 3(\tfrac{3}{2}) \oplus 3(-\tfrac{1}{3}) \oplus 3(-\tfrac{1}{3}) \oplus \bar{3}(-\tfrac{2}{3}) \oplus \bar{3}(\tfrac{1}{3}) \oplus \bar{3}(\tfrac{1}{3})$
$s=½: 56 → 1(½) \oplus 1(-½) \oplus 6(-\tfrac{1}{6}) \oplus \bar{6}(\tfrac{1}{6}) \oplus 1(-½) \oplus 1(½) \oplus 8(½) \oplus 8(-½) \oplus 3(\tfrac{1}{6}) \oplus 3(-\tfrac{5}{6}) \oplus 3(\tfrac{1}{6}) \oplus \bar{3}(-\tfrac{1}{6}) \oplus \bar{3}(\tfrac{5}{6}) \oplus \bar{3}(-\tfrac{1}{6}) \oplus [3(\tfrac{1}{6}) \oplus \bar{3}(-\tfrac{1}{6})]$
$s=0: 70 → 1(0) \oplus 1(0) \oplus 1(1) \oplus 1(0) \oplus 1(-1) \oplus 8(0) \oplus 8(0) \oplus 3(-\tfrac{1}{3}) \oplus \bar{3}(\tfrac{1}{3}) \oplus 6(\tfrac{1}{3}) \oplus 6(-\tfrac{2}{3}) \oplus \bar{6}(-\tfrac{1}{3}) \oplus \bar{6}(\tfrac{2}{3}) \oplus [3(\tfrac{2}{3}) \oplus 3(-\tfrac{1}{3}) \oplus 3(-\tfrac{1}{3}) \oplus \bar{3}(-\tfrac{2}{3}) \oplus \bar{3}(\tfrac{1}{3}) \oplus \bar{3}(\tfrac{1}{3}) \oplus 1(0)]$
The number in the round brackets after each SU(3) representation is the hypercharge, Y, of that representation. The SU(3) representations in square brackets in the decomposition of the 56 and 70 are the goldstino and Goldstone modes respectively, and will therefore be eaten.
In Gell-Mann's proposal, up-type quarks come from 1 + 8, down-type quarks come from 6 + 3bar or 6bar + 3, and leptons come from 3 or 3bar. The leptons are definitely in the same N=2 supermultiplet as some of the triplet gravitinos. The other triplet gravitinos are paired up with the "3" part of the down-type quarks. Meanwhile, there is no immediate relationship between the singlet gravitinos and the "1" part of the up-type quarks, since the gravitinos are in the graviton supermultiplet, whereas the "1" part of the up-type quarks belongs instead to a SU(3)-singlet vector multiplet extending down to the scalars.

By this analysis, the relationship appears to be the reverse of what I've suggested: the triplet gravitinos are more tightly linked to the leptons than to the quarks. However, that's just an N=2 relationship; the rest of the full N=8 algebra is hidden here. Ultimately everything is connected to everything else, since it's all in the same supermultiplet; but it's a question of finding how those relationships translate into physics.

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arivero
Gold Member
arivero
Gold Member
Not sure about the U(1) charges, so lets just to see how a N=1 preserving SU(3) could match.

1

1 + 1 + 3 + 3

1 + 8 + 1 + 3 + 3 + 3 + 3 +3 + 3

1 + 8 + 1 + 8 + 6 + 3+ 6 + 3 +3 + 3 +3 +3 + [3 + 3]+ 1 + 1

1 + 1 + 1 + 1 + 1 + 8 + 8 + 3 +3 + 6 + 6 + 6 +6 + [3 + 3 + 3 + 3 + 3 + 3 +1]

See any pattern here? At least, it says that one Weyl-like u,c,t nonet has spin 1 superpartners instead of spin zero.
And there are two extra spin 1/2 singlets. As for the triplets, I am in doubt: they could be the leptons, they could be
usual gauginos, they could be the triplets of the down sector, or even perhaps they could be goldstinos.

In the spin 1 side, I would like to find some reflect of the fact that SU(8) has 28 generators while SU(3)xSU(2)xSU(2) has 14.
Code:
     O              O
/
O--O           O--O
\
O              O
But the solitary boson with a 3/2 partner could hint to B-L instead of EM, and then to U(1)xSU(3)xSU(2)xSU(2), with 15 generators.

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In copying out the decomposition, I reproduced an error from the paper: the spin-1 boson with hypercharge "3/2" should have hypercharge 2/3.

Marni Sheppeard has posted http://pseudomonad.blogspot.com/2011/11/koide-quarks-ii.html" [Broken], and an interpretation in terms of her framework will be forthcoming. What I propose to talk about here is how this structure might fit into the more orthodox field-theoretic framework described in this thread.

But perhaps I should begin by mentioning some new details. We are looking at a critical point of N=8 supergravity, which Nicolai and Warner in 1985 calculated to possess the features of Murray Gell-Mann's 1983 phenomenological proposal. This is a critical point of N=8 supergravity in anti-de-Sitter space, so we already know that for phenomenology, it will need to be uplifted to de Sitter space somehow.

In recent years, this critical point has been studied as an instance of the http://arxiv.org/abs/1012.3999" [Broken], it would not surprise me to learn that the Gell-Mann scenario is realized, but at a different point. However, Scenario I has the utility of being a gauge/gravity dual where both sides are known, so it can be studied closely.

In the Koide thread linked above, Alejandro pointed out a famous 1979 paper by http://dx.doi.org/10.1016/0370-2693(79)90842-6" [Broken] (600+ cites), in which the idea is introduced that at the GUT scale, the bottom mass equal the tauon mass, the muon mass equals three times the strange mass, and the down mass equals three times the electron mass. They even provide a Higgs construction in which these relations result from the existence of three colors (and further assumptions about a hierarchy of masses). Since Alejandro's new strange-charm-bottom / e-mu-tau relation also involves a factor of three, and since we are also looking at a 3:1 ration of dark energy to dark matter, the logical thing to do is to try to explain all of these as resulting from the existence of 3 colors - or 3 generations, or some combination of the two (recall that Gell-Mann's SU(3) is the diagonal subgroup of the product of color SU(3) and family SU(3)).

The first step would be a unified explanation of the Georgi-Jarlskog relations and the extended Koide relations. I suppose the 70 N=8 scalars would be the Higgses - or, in Yoshio Koide's terminology, the "yukawaons", whose expectation values contribute to the Yukawa couplings. I note that one of http://adsabs.harvard.edu/abs/1990MPLA....5.2319K" does actually explain his formula using an 8+1 nonet of scalars, and such a nonet is available in the N=8 scenarios under consideration! Also note that he talks about a U(3) family symmetry. So it seems possible that this could all work out. Of course it would be miraculous if it did. But at least we are incrementally approaching the point where it will be possible to do more than point out suggestive possibilities.

Now I want to say something about the cosmological aspect. In this thread I've talked of explaining dark matter and dark energy in terms of gravitinos, and certainly gravitinos are a standard candidate for dark matter. I haven't seen any "gravitino dark energy" work, but there are papers about gravitino condensates, so one would think in terms of a condensate with the necessary equation of state.

However, the starting point of all this was Louise Riofrio's cosmology. This is a varying-speed-of-light (VSL) cosmology which I think can be understood by reinterpreting the "Milne model" of an empty universe, in terms of VSL. I thought I had a link handy which could explain this, but can't find it... Anyway, I think it works by saying that light slows down at a certain rate, and also that the critical density is maintained (this appears to be an axiom), so there is a 1-3/pi deficit in the energy density. Then there is a geometric argument for the 3:1 split of the deficit into dark energy and dark matter - except that dark energy is not an energy, it's just the non-material part of the deficit. Or, something like that; here is her http://www-conf.slac.stanford.edu/einstein/talks/aspauthor2004_3.pdf" [Broken].

I mention this, not just to remind the reader of where this concept started, but because one should expect that anything to do with gravitinos has a dual interpretation which is "geometric" rather than "material". In fact, this is an opportune moment to mention another aspect of Sheppeard's synthesis, the braid representation of the standard model fermions. (There is another representation she uses, in terms of nonassociative paths through a "tetractys".)

In AdS/CFT, fields in the bulk are dual to traces of cyclic products of operators in the boundary theory - see the AdS4/CFT3 review by Klose, linked above. The boundary fields are thus "holographic preons" for the bulk fields. It seems to me that if Sheppeard's braids and paths have some relationship to a dS4/CFT3 implementation of the standard model, such as an uplift of one of these N=8 critical points, it will have something to do with these cyclic traces. And indeed, she talks of the mirror neutrinos as a holographic emanation from the cosmological horizon, so again we have a suggestive compatibility.

Now here I want to mention last week's paper by http://arxiv.org/abs/1111.2361" [Broken], in which the neutrino mass scale is set by the size of the cosmological constant, in a theory with broken R-symmetry. R-symmetry is the "symmetry of supersymmetries" possessed by a theory with extended supersymmetry. In gauged N=8 supergravity, the R-symmetry is SO(8), rotation among the eight supersymmetries; the U(1) at the critical point discussion is the remnant of this R-symmetry.

So what I want to point out is the possibility of a holographic-geometric realization of the Davies-McCullough relation. If through some miracle it turns out that an uplift of the Gell-Mann-Nicolai-Warner... vacuum is the same thing as the Riofrio-Sheppeard cosmology, one might expect that that is part of how it works.

Finally, I can't let the still-unvanquished faster-than-light neutrinos go unremarked. There are many, many explanations of the FTL neutrinos in circulation now, and in terms of the current scenario, one might think of using extra dimensions (since d=4 N=8 supergravity can be obtained as a compactification of d=11 M-theory). However, I'll also mention my https://www.physicsforums.com/showthread.php?t=535480#6" that there might be gravitino-mediated "superoscillations" between neutrinos and tachyonic scalar superpartners. That idea has problems (follow the link), but if it does make sense, once again, some of the many N=8 scalars may be able to fill this role.

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                                motivic exceptionology
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idempotent calculus -- categorified twistor polytopes -- new cosmology
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twistor supergravity
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yukawaon models -- de Sitter uplift of "Scenario II" -- gravitino dark sector
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"Scenario I" (AdS/CFT)
That sums up much of what I've talked about in this thread.

The top half - everything above "twistor supergravity" - is Sheppeard's synthesis. The "idempotent calculus" refers to Carl Brannen's formalism, the "new cosmology" refers to Louise Riofrio's work, and by "motivic exceptionology" I mean the idea that the laws of physics result from some exceptional algebraic structure (exceptional in the sense of A-D-E classifications). Along with Sheppeard, Michael Rios contributed to that part of the web of ideas.

The bottom half contains the essentially orthodox framework that I cooked up or stumbled upon in an attempt to translate the top half. "Yukawaon models" are the work of Yoshio Koide, "Scenario I" and "Scenario II" are scenarios for N=8 supergravity discussed earlier in this thread (but only Scenario I has been shown to make sense), and "gravitino dark sector" refers to the idea of getting dark energy as well as dark matter from the multiple gravitinos of an extended supergravity.

Twistor supergravity is a real subject that many people are working on, and it provides the clearest bridge between top and bottom.

I don't claim that this little mandala even depicts a working schema for a theory of everything; it's more a map of a concept space that can be explored.

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The particle masses and mixing matrices contain so many unexplained details, and there have been so many false alarms of new physics this year, that I was beginning to think I would be paying no attention at all to news and rumors from the colliders. But there is a new preprint, harking back to a report 3 years ago of "lepton jets" - muons showing up in pairs, at a distance from the site of collision, as if they were being produced by the decay of an unknown particle (which would form in the collision, travel a short distance, and then decay). As the second link above mentions, a few people from the CDF collaboration tried to explain the lepton jets by positing new intermediate states equal to 2, 4, and 8 times a tauon mass; and a few weeks earlier there had been a theory paper predicting lepton jets as a result of broken symmetries in the dark sector, though not with the peculiar and prominent signatures claimed by the CDF.

In this thread, I've been touting the idea of multiple gravitinos, and the decay of dark matter gravitinos has been much discussed as a possible source of the PAMELA positron anomaly. So I thought I would at least log the possibility that the CDF multimuon anomaly might have something to do with these gravitinos (or mirror neutrinos, if you prefer). My notion has been that the N=8 gravitinos all have the same mass and the same abundance, but the triplet gravitinos have a dark-energy equation of state. Meanwhile, the tauon mass features in that impressive piece of numerology (the Koide relation) which seems to provide a template for understanding all the fermion masses, and here we have intermediate states consisting of the tauon masses times powers of two (recall that N=2,4,8 are the most interesting forms of extended supersymmetry). So I don't know how it would work, but it's surely worth mentioning.

Also, if you go back to the start of this thread, there's a quote from Hermann Nicolai, regarding Gell-Mann's N=8 scenario (dubbed "Scenario II" by Klebanov and co-workers, the proposal for the SU(3) x U(1) fixed point that they could not realize via AdS/CFT), in which he speculates that QCD color-flavor locking might have something to do with the appearance of the diagonal subgroup of SU(3)color x SU(3)family in Gell-Mann's construction. Color-flavor locking occurs in a high-density phase of QCD with three light quarks - in the real world that's up, down, strange. Stanley Brodsky has proposed that the usual QCD condensates are confined within the hadrons, rather than permeating space as normally supposed, and that this should eliminate the QCD contribution to the cosmological constant. (Zhitnitsky, meanwhile, wants to get dark energy from QCD.)

I am led to wonder about two things. First, if the QCD condensates are confined inside the hadron, as in the bag model, could the space inside a hadron be in the CFL phase, at least some of the time? (In the CFL phase, there are ud, us, and ds diquark condensates; other regions of the QCD phase diagram might only contain one or two of these diquark condensates.) Second, some of the extended Koide relations involve a light-quark triplet (uds) and a heavy-quark triplet (cbt). In Alejandro Rivero's scb triplet, the formula for which has a remarkable similarity to Koide's original e-mu-tau triplet, the square root of the strange quark mass enters with a different sign. I'm wondering if the light quarks should all be regarded as having this "different sign", compared to the heavy quarks, and if this has something to do with their involvement in CFL. (Koide formulae involve square roots of masses; or, looked at alternatively, they involve quantities which have to be squared to produce the mass. From this second perspective, there's no problem about the quantity being negative. For example, it might be a VEV, as in Koide's yukawaon models; it might indicate that the fermion mass term involves a bilinear coupling to the field with the VEV.)

arivero
Gold Member
Let me stress that the different sign allows for quasiorthogonality (exact in the limit where the phase of leptons is 15 degrees instead of 12.5 (or 0.2222)) between the quar triplet and the lepton triplet. In your suggestion, you could be telling that leptons are there, orthogonal to s-c-b and not, say, c-b-t, to make use of the change of sign. The only problem is that a similar triplet with -u (or -d), -s and +c does not seem to exist. Well, empirical search has limits, because of that we do models.

Something special about the structure of the s-c-b triplet is that it has two quarks with charge -1/3 and one quark with charge +2/3. This stands out for me, after reading about CFL, because it shares this feature with u,d,s, and that the charges sum to zero is part of what makes CFL work. So I suggest that what comes first is that e-mu-tau is related to such a quark triplet, and then the quark masses arrange themselves around this. But we should discuss this in the Koide thread.

arivero
Gold Member
Yep, for this thread perhaps the only extant remark is that initial work on these mass relationships was heavily based on breaking SU(2)_L x SU(2)_R down to SU(2)xU(1). This is probably relevant for N=8 but I am not conversant on the fixed points of the theory. What is true is that both groups have a link with kaluza klein: the former is, disguised as SO(4), the isometry group of the sphere S3, the latter is the isometry group the product of spheres S2xS1.

This year-old thread contains a confused melange of ideas and perhaps I should want to keep it buried in the archives. But recently I found a line of inquiry arising from it that seems worth recording somewhere.

First, here is a summary of the more defensible ideas.

1) Nicolai and Warner studied a particular ground state of N=8 supergravity with N=2 supersymmetry and SU(3) x U(1) gauge symmetry. In the early 1980s, Gell-Mann tried to get the SM fermions out of this construction - or at least their quantum numbers. The masses and the geometry (AdS) were definitely wrong. (But Nicolai, at least, still harbors hopes that it might somehow be relevant to reality.)

2) In the 2000s, the Nicolai-Warner theory was studied again, not for phenomenological purposes, but as an example of AdS/CFT. To work, this requires a slight revision of the Nicolai-Warner scheme, to the "Scenario I" mentioned in comment #17.

3) The theory has six gravitinos transforming as (anti)triplets of the SU(3) symmetry, and two gravitinos which are SU(3) singlets. My "idea" was that the dark energy is a condensate of the triplet gravitinos, and the dark matter is made of the singlet gravitinos. This is motivated by the fact that the dark energy's fraction of the cosmic energy density is about three times that of the dark matter. But to obtain that conclusion, you seem to need the further premise that the various species of gravitinos have roughly equal mass and abundance; which is possible but hardly inevitable.

Summed up like that, the idea may seem simple and logical enough to be worth considering. It does have a major weak point, which is the contrived nature of the attempt to get the SM fermions from the spin 1/2 fields. It never worked very well and everyone dumped it after the 1984 superstring revolution.

However, something I do like is that a genuine AdS/CFT duality is part of the theoretical mix here. AdS/CFT is deep and powerful, one should expect that the real world works on similarly deep principles, and so one shouldn't throw away an opportunity to connect the two. i was aware that one should be interested in "uplifting" this duality to dS/CFT to make it phenomenologically relevant, but I wasn't sure how to do that. We still only have just one semi-functional example of dS/CFT, and the boundary theory there looks rather different to the boundary theory of its AdS/CFT counterpart.

However, it has occurred to me that the bulk theories are the same for both (a purely bosonic Vasiliev gravity), except for the cosmological constant. So there is a "correspondence of correspondences", in which you can take the bulk fields, and look at their boundary counterparts in AdS and dS respectively. And you could do the same for the Nicolai-Warner theory! We don't know what the future CFT operators for bulk fields of "Nicolai-Warner in dS space" are, but perhaps you can reason about them, using (i) knowledge of their AdS counterparts, and (ii) how the "correspondence of correspondences" works for the simpler Vasiliev theory mentioned above.

Of course, we don't live in de Sitter space. But we may live in a universe transitioning from one dS steady state to another. This is the basis of Penrose's new cyclic cosmology, it shows up in Padmanabhan's recent attempt to calculate the cosmological constant, and it was the scenario in a paper by Andrew Strominger called "Inflation and the dS/CFT Correspondence". Strominger didn't have a concrete example of a cosmic boundary CFT to work with (that was still ten years away), but he discussed some of the properties such a CFT would need to have, in order to be dual to a universe that started in dS and ended in dS. Clearly these considerations would be something to bear in mind, when studying the possibility of a dS uplift of Nicolai-Warner.

One final ingredient which I think is worthy of mention. Usually it's a long road from the fundamental parameters of particle physics to the observable parameters of cosmology. Idea 3 above is unusual in proposing that the DE/DM ratio follows directly from the ratio "number of triplet gravitinos : number of singlet gravitinos" (as noted, it requires the additional premise of mass and abundance degeneracies). So I'll point out the obscure work of Arthur Chernin, who wants to use an extended version of the cosmic coincidence problem to deduce fundamental properties of particle physics with a similar directness. The concept then becomes one of realizing the N=8 gravitino cosmology through a Nicolai-Warner uplift in a Chernin-Strominger framework. It is at least something to work on.

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