The wrong turn of string theory: our world is SUSY at low energies


Gold Member
The 54 down to [itex]SU(5) \times U_1(1)[/itex]
54 = {15}(4) + \bar{15}(−4) + {24} (0)
[/tex]Then each representation goes down to [itex]SU(3) \times SU(2) \times U_2(1)[/itex]
15 =& (1, 3)(−6) + (3, 2)(−1) + (6, 1)(4) \\
24 =& (1, 1)(0) + (1, 3)(0) + (3, 2)(5) + (\bar 3, 2)(−5) + (8, 1)(0)
There is no shortage of theoretical options to investigate. But I now prefer to think of quark-diquark supersymmetry as something which manifests only at the end of an open string,
I am doing a fast review of the bibliography; I'd say we have accumulated a lot. The main problem with the open string formulation is that there are two simultaneous bootstraps putting charges at the ends of the string: the one of the generation group, via supersymmetry, and the one of su(3), via (3 x 3)_anti = 3. They can not be independent because the 15 of SU(5) goes with the 3 of SU(3), whule the 24 goes with the singlet. Which amazingly could be compatible with a claim (hep-ph/9606467) that 54 and higher representations of SO(10) are always in the singlet of any other factors.

Half a 54, which we can do because it contains both particles and antiparticles, is a 27, and then the search scope becomes too wide. A traditional mention is (3,3,3) + (/3,/3,/3) of SU(3)^3, falling from E6 (but it is more typical to smash it into 16+10+1 of SO(10). In both cases, further breaking is needed if we want to get something close to the above decomposition)

Other 54 pathway, which can appear from branes too, is from the 55 of Sp(10), with only the nuissance of the extra singlet. The appendix of hep-th/0305069 mentions that this 55 could be obtained from orientifolds, but it doesn't give a reference. 1206.0819v2 suggest a realization with D7-branes. hep-th/0204023 Uses D7, D3 and O3 for generic Sp(2N+2M)xSp(2N), but does not evaluate our particular N=3 M=2. Neither do Luty et al hep-th/9603034v2 when looking at Sp(2N) susy. Similarly Witten 83 go for generic Sp(2N). This is an interesting paper even if if it focuses on its use as coloration.

A recent work arXiv:1603.05774v2 considers the "hidden pions" in the 15 both comming from SO(10) and Sp(10), and it proposes mass formulae! I do not get how it presents them as pions and not diquarks.

Also recently arXiv:1608.01675v1, Arkani-Hamed et al, mentions the decomposition from 15 into SU(5) with quantum numbers from the standard model. Comparing with this one, and with Vachaspati, it seems that the innovation here in this thread is the use of the most internal SU(3) for flavour instead of colour; the logic being that the string will take care of colour.

Googling for group decompositions, even with flag, is very inneficient, so sure I am missing important references.
Last edited:


Gold Member
it seems that the innovation here in this thread is the use of the most internal SU(3) for flavour instead of colour; the logic being that the string will take care of colour.
I would like to find some strong -pun intended- argument to use SU(3) instead of any other SU(N), big N etc... Lacking this, it could be worthwhile to note that having a 54 and pasting an U(3) upon it is pretty interesting. It could be nice to have Lisi or Toni Smith or some other big numerologists here in the thread. On my side, lets at least notice that
[tex]54*9 = 496-1-9 [/tex]

\\ =45+90+45+90+192+24+8+1\\ =495[/tex]

If this is a branching rule of something into something, I do not know. It looks so, but I am not conversant with large representations.

EDIT: from wikipedia–Schwarz_mechanism
Green describes finding 496 on each side of the equals sign during a stormy night filled with lightning, and fondly recalls joking that "the gods are trying to prevent us from completing this calculation".
A more modern analysis is quoted by Lubos here but no clue of where our representations comes from.
Last edited:


Gold Member
Actually I was thinking other two different venues to approach 496:
  1. somehow expand SO(32) as a sum "SO(1+5+10+10+5+1) " and consider only the 10 for particle content. This way could be useful if colour is not invited to the party
  2. promote each preon with a colour label, so the matrix is promoted from dim 10 to dim 30. Still, we need to add manually an extra preon/antipreon pair, uncolored, to get up so SO(32). This looks more natural that the previous post where we only have dim 30 and it is the 3x3 submatrices that overfill across the diagonal fo fill up to 495 "states".
Then, of course, both cases -and the previous one- need some argument to extract only the sm-like representations.
Last edited:


Gold Member
I am starting to suspect that special embeddings where not fully explored in the old age. Witten 84 does SO(32) to SU(5) via the most trivial way, times SO(22) and then not looking for generations nor colour (as it is already considered in the GUT group). Fortunately Gell-Mann, Ramond and Slansky look for colour too... but, uh, do they forgot the [itex](1, n^2_3 -1, 8^c)[/itex] here?


Formula 2.18 for our case should decompose SO(32) as [tex]
(n \times n)_A= [1,1,1^c] + {\bf (1,24,1^c) }+ (1,1,1^c)
+(1,1,8^c) + (2,5,3^c) + (2,\bar 5, \bar 3^c)
+{\bf [1,15,\bar3^c]} + {\bf [1, \bar {15}, 3^c]}
+[1,10,6^c] + [1, \bar 10, \bar 6^c]
+ 1,24,8^c

For verification, it is possible to branch down to this same result using the new tables of via regular branching to [itex]so30[/itex] and [itex]su15[/itex], and then special branching down to [itex]su15 \times su3[/itex].
  1. For so30 ⊕ u1(R):
    • 496 = (435)(0) ⊕ (30)(2) ⊕ (30)(−2) ⊕ (1)(0)
  2. Then for su15 ⊕ u1(R):
    • 435 = (224)(0) ⊕ (105)(4) ⊕ (105)(−4) ⊕ (1)(0)
    • 30 = (15)(2) ⊕ (15)(−2)
  3. and then ⊃ su5 ⊕ su3(S):
    • 224 = (24, 8) ⊕ (24, 1) ⊕ (1, 8)
    • 105 = (15, 3) ⊕ (10, 6)
    • 15 = (5, 3)
    • (and none from 120 = (15, 6) ⊕ (10, 3) )
Perhaps some way down via so12⊕ su3(S), and so12 is so(2)xso(10) and then so(2)xsu(5)? Also, perhaps O(n) and U(n) instead of su, so?

I am not sure if this is the right branching, or it is the former one, or
some other, but I find a bit disappointing that the final conclusion
of this thread is going to be to identify the lost scalar partners as bosons of the SO(32) string. Such
idea should already be in the literature somewhere. :oops::sorry:
Last edited:
I have been worried all week that you are too optimistically jumping between flavor symmetries and gauge symmetries. For example, that scalar 54 which is supposed to come from the mesons and scalar diquarks of five quark flavors, assumes an SO(10) flavor symmetry, which is rather unusual. But in most of this week's rampage through representation theory, you've been looking at gauge groups, not flavor groups.

As you say, Gell-Mann et al is good because they are looking at flavor and color together. So we have at least one clear example of how to do that. But there are further twists. When the theory is strongly coupled, there may be chiral symmetry breaking that reduces the flavor group, and determining that is an art in itself. (It's very very likely that this is related to the sBootstrap, since the pions are precisely the Goldstone bosons of chiral symmetry breaking.)

Also, "there are no global symmetries in string theory". There is a worldsheet theorem that if a global symmetry exists, there must be a corresponding gauge-boson state of the string, turning it into a local symmetry. In Sakai and Sugimoto's holographic QCD, flavor is gauged. But it's also possible for the global symmetry to just be approximate. There is some discussion here.
I have just run across two highly relevant papers by Armoni - 1310.2027 and 1310.3653. They came out near the start of this thread's long dormant period, from late 2013 through all of 2014... The first one, in particular, is remarkable for how many of our themes it contains.

I'll set the scene with a remark from that first paper (page 7). We are dealing with a field theory which is realized in string theory by a "Hanany-Witten" brane configuration "identical to the brane configuration that realizes SO(2N) SQCD, except that the D4-branes are replaced by anti D4-branes".

I'm emphasizing this because, if we do have to study this one in detail, we know that the place to begin is with the configuration that realizes SO(2N) SQCD. Armoni is interested in a similar but non-supersymmetric theory; but it may be that we will want to go back to the supersymmetric prototype.

Another thing to note is that these Hanany-Witten configurations can be lifted to M-theory. In Type II theories, they appear as a web of D-branes and NS-branes (and in this case, an orientifold plane), but in M-theory, they can be realized as a single M5-brane, on the right geometric background.

Armoni is concerned with two field theories, an electric theory and a magnetic theory. He is proposing a Seiberg duality. Inter-brane forces which cause the branes to rearrange themselves are also a part of it.

What I want to note here, are the symmetries and some of the particle content. The flavor symmetry is SO(2Nf). There are particles in the non-supersymmetric theory (but which is, remember, descended from a supersymmetric theory) which he calls quarks, squarks, a gluino, a meson, and a mesino. The gluino transforms in an antisymmetric two-index representation of the gauge group, so it might be a toss-up as to whether the gluino or the squarks are more like diquarks.

On page 18, the breaking chain SU(2Nf) -> SO(2Nf) -> U(Nf) is referenced. And the companion paper talks about chiral symmetry breaking.


Gold Member
The next temptation is to try to classify under chirality those bosons we have got in a 496 (or 495 or whatever), or the original 54 ones. If we assume that they were produced, in SO(10), via a set of five "quark preons" and "five antiquarks", then the next step is again obvious: consider sum and difference of particle-antiparticle, as such is the way to build chirality invariant states, and see what happens with the group? Does it decompose to a product of two groups, one for left, other for right chiral? And when we scale up, what does happen? Does 496 divides in 248+248, or 495 in 1+247+247 ?

(amusingly #247 is the number of this post in the thread... yes, numerology is always a running joke here :-)
Last edited:
Today I found some old research which might help with the bottom-up issues. In the 1980s, Masaki Yasue wrote papers on obtaining SM fermions as goldstone fermions in SQCD. With Terazawa, he found some mass formulae based on Dashen's formula. Then in the 1990s, Craig D. Roberts studied a common Dyson-Schwinger approach to meson and diquark propagators, form factors, etc. He wrote specifically about Goldstone's theorem and also the Pauli-Gursey symmetry. He doesn't mention supersymmetry, but Dyson-Schwinger equations can be extended to superfields.
Something that bothered me about the recent group-theoretic explorations, is that they didn't take the goldstone nature of pions into account. They were trying to get the SM from open strings, but pions aren't just strings, they are strings that emerge from chiral symmetry breaking.

Meanwhile, Mizoguchi and Yata explicitly talk about deriving the SM from goldstone fermions in string theory, from the spontaneous breaking of some geometric symmetry. They are implementing "coset sigma models", and they say, "The advantage of the coset sigma model approach is that the associated quasi-Nambu-Goldstone fermions are typically chiral."

It also turns out that QCD's chiral symmetry breaking can be understood in these terms (i.e. as a coset sigma model). This encourages me to think that all of this can come together, that the SM really might be obtained from the goldstone fermions of some stringy SQCD.

I even wonder if we could find a stringy construction which has E8L x E8R chiral symmetry, and in which a "496" decomposition is implemented by chiral symmetry breaking.
"Quasi Nambu-Goldstone Fermions" (Buchmuller et al, 1983) is one of the fundamental papers on this topic. They describe how, in passing from a coset sigma model to its supersymmetric counterpart, the symmetry group is complexified, doubling the real coordinates of the coset space, and adding to the original Goldstone bosons, a set of "quasi Goldstone bosons". Taken together, these are the superpartners of the goldstone fermions.

The Goldstone scalars form the coordinates of a geometric space, a Kahler manifold. For the nonsupersymmetric case, the metric of this Kahler manifold is unique, and it uniquely determines the sigma-model lagrangian. But for the supersymmetric case, the quasi Goldstones double the coordinates of the geometry, and away from the "Goldstone hyperplane" the metric - and consequently the lagrangian - is no longer unique.

These sigma models are effective theories. The parts of the sigma-model lagrangian that are not determined by the coset geometry, are determined by details of the deeper theory that has undergone spontaneous symmetry breaking. For example, suppose we had a brane stack in a compactification, with some strongly coupled supersymmetric theory as its worldvolume theory. The basic properties of the brane stack may imply a particular coset sigma model as effective theory, while the geometric details of the compactification may determine the details of the lagrangian.

In terms of the sbootstrap, one could then proceed as follows. Identify a sbootstrap supersymmetric sigma model, such that the SM fermions are its goldstone fermions; and perhaps a specific potential for the fermion masses. Then find a brane configuration which implements that sigma model, and a compactification geometry which induces the desired potential. This 2016 paper offers a small start by considering possible supersymmetric mass terms for pions.
A small observation:

The sbootstrap has two parts: a version of the traditional quark-diquark "hadronic supersymmetry", and an extension to include a lepton-meson supersymmetry.

These supersymmetric coset models can actually give us the latter, e.g. see the paper in #222. And they can also give the full SM fermion spectrum, e.g. Kugo-Yanagida.

But in Kugo-Yanagida, there is nothing like quark-diquark supersymmetry. Quarks and leptons are on the same level, whereas I would look for quark-diquark supersymmetry to be realized at a deeper level (perhaps as in Brodsky et al).

That would mean that quarks are fundamental, while leptons are super-composites, as in #222 (though the phenomenological quarks may also be super-composites). #222 offers an SQCD implementation of this for Dirac fermions.

But what about the chiral fermions of the full SM? In #239 and #241, @arivero described an N=1 SU(5) theory. Superficially, it's just another GUT; but it's supposed to provide the context for a chiral implementation of the sbootstrap.

My small observation is just that, if we stick with the idea that the quark superfields are fundamental and the lepton superfields are composite... the leptons all come from the 24. So that would mean that, for a chiral sbootstrap theory, the 15 and 15-bar would be what's fundamental.
Last edited:


Gold Member
By the way, Mitchel, have you already commented on Mourad-Sagnotti here in this thread? Interestingly, the most cited paper on USp(32) seems to be from Sugimoto, the one of the Sakai-Sukimoto model. It would seem that the open-strings guys are into something.


Gold Member
Branching down from USp(32) the main difference is that the "padding" (the dim 30 irreps that are foreign to our construct but complement the also extra (10,6) irreps of su5 + su3 ) gets itself arranged as a SU(2) doublet instead of oppositely U(1)-charged objects.
  1. usp32 to su2 ⊕ usp30
    • 32 = (2,1) + (1,30)
    • 495 = (2,30)+ (1,434)+(1,1)
    • 528 = (3,1) + (2,30) + (1,465)
  2. And then down to su15 ⊕ u1(R):
    • 434 = (224)(0) ⊕ (105)(2) ⊕ (\bar 105)(−2)
    • 30 = (15)(1) ⊕ (\bar 15)(−1)
And then su15 ⊃ su5 ⊕ su3(S) proceeds in the same way that above

Objection here is that the 495 in the USp(32) string is only for fermions, not for scalars. Also, once USp is in the game, one should consider the ""dual" SO(33), should we? On the other hand, we could have considered to start the game from SO(11) instead of SO(10) or SU(5). It seems that it adds more complexity without further physics.
I don't think that I've ever seen a symplectic algebra in GUT model building. I decided to look at the algebra breakdown in more detail, using my SemisimpleLieAlgebras Mathematica notebook that I'd written. For SU(5) model building, one wants 1, 5, 10, 10*, 5*, 1 for the elementary fermions, 24 for the gauge fields, and 5, 5* for the Standard-Model Higgs particle. We also Higgs-EF interactions H(5).F(1).F(5*), H(5).F(10).F(10), H(5*).F(5*).F(10), and Higgs-Higgs interaction H(5).H(5).

Symplectic-algebra irreps:
Vector: {1,0,0,...}, Antisymmetric 2-tensor: {0,1,0,...}, Symmetric 2-tensor, adjoint: {2,0,0,...}
The antisymmetric one is sort of traceless, with the algebra's antisymmetric form subtracted out.

Unitary-algebra irreps:
Vector: {1,0,0,...}, Vector conjugate: {0,...,0,0,1}, Antisymmetric 2-tensor: {0,1,0,...}, Antisymmetric conjugate 2-tensor: {0,...,0,1,0}, Symmetric 2-tensor: {2,0,0,...}, Symmetric conjugate 2-tensor: {0,...,0,0,2}, Adjoint: {1,0,0,...,0,0,1} ((vector * conjugate vector) - scalar)

The first one is from extension splitting, as I call it: Sp(2(m+n)) -> Sp(2m) * Sp(2n)
Sp(32) -> Sp(2) * Sp(30) -- with Sp(2) ~ SU(2)

The second one is what I call root demotion, with the long root reduced to a U(1) factor: Sp(2n) -> SU(n) * U(1)
Sp(30) -> SU(15) * U(1)

The third one is a Kronecker-product decomposition: SU(m*n) -> SU(m) * SU(n)
This is a sort of outer product with original first indices flattened, and original second indices flattened. It is done on the fundamental rep.
SU(15) -> SU(5) * SU(3)

I'll now decompose the original irreps:
Vector: 32 = (2,1) + (1,30)
= (2,1,0) + (1,15,1/2) + (1,15*,-1/2)
= (2,1,1,0) + (1,5,3,1/2) + (1,5*,3*,-1/2)
Antisymmetric: 495 = (2,30) + (1,434) + (1,1)
= (2,15,1/2) + (2,15*,-1/2) + (1,105,1) + (1,105*,-1) + (1,224,0) + (1,1,0)
= (2,5,3,1/2) + (2,5*,3*,-1/2) + (10,6,1) + (10*,6*,-1) + (15,3*,1) + (15*,3,-1) + (24,8,0) + (24,1,0) + (1,8,0) + (1,1,0)
Symmetric: 528 = (3,1) + (2,30) + (1,465)
= (3,1,0) + (2,15,1/2) + (2,15*,-1/2) + (1,120,1) + (1,120*,-1) + (1,224,0) + (1,1,0)
= (3,1,1,0) + (2,5,3,1/2) + (2,5*,3*,-1/2) + (15,6,1) + (15*,6*,-1) + (10,3*,1) + (10*,3,-1) + (24,8,0) + (24,1,0) + (1,8,0) + (1,1,0)

One can get singlets for both the 5-5* and the 5*-5*-10 SU(5)-model interactions. But it does not seem to be possible for the 5-10-10 one. This model also seems to lack right-handed neutrinos.
Oops, some typos. The (MS)SM Higgs interaction is supposed to be H(5).H(5*). I also omitted some initial 1's in the antisymmetric and symmetric 2-tensor results:
AntiSym = (2,5,3,1/2) + (2,5*,3*,-1/2) + (1,10,6,1) + (1,10*,6*,-1) + (1,15,3*,1) + (1,15*,3,-1) + (1,24,8,0) + (1,24,1,0) + (1,1,8,0) + (1,1,1,0)
Sym = (3,1,1,0) + (2,5,3,1/2) + (2,5*,3*,-1/2) + (15,6,1) + (1,15*,6*,-1) + (1,10,3*,1) + (1,10*,3,-1) + (1,24,8,0) + (1,24,1,0) + (1,1,8,0) + (1,1,1,0)

The tensor ones have SU(5) irreps in addition to what appears in the Standard Model: 15 and 15* (symmetric 2-tensor and its conjugate).
5* -> (1,2,-1/2) and (3*,1,1/3) -- L lepton, R down quark
10 -> (3,2,1/6) and (3*,1,-2/3) and (1,1,1) - L quark, R up quark, R electron
15 -> (3,2,1/6) and (6,1,-2/3) and (1,3,1) - L quark, R up quark with QCD multiplet 6* instead of 3, R electrons with charges 0, -1, and -2
Thus making some elementary fermions that we do not observe.


Gold Member
Have you uploaded/published your package somewhere? I have always found intriguing that Wolfram has not a standard package for this task.

AntiSym = ... + (1,15,3*,1) + (1,15*,3,-1) + ... + (1,24,1,0)
Thus making some elementary fermions that we do not observe.
To me, this (1,15,3*,1) and the colour singlet (1,24,1,0) are the real meat that should survive at low energy. They contain three (or six, which could be unsurprising if mirrors are required) generations of something. Also, the last U(1) charge is proportional to "baryon number" so it can be used to align leptons and quarks.


Gold Member
The big thing of the 15 as it goes down to three "pairs of generations" is that while we can not see the electroweak force, we can see that if it appears it is going to be chiral: each "pair of generations" has two -1/3, two +2/3 and one 4/3 quark. If we want to suppress the latter at low energy, we need a combination of SU(2) unable to see it.

Note that during this thread I was more in the side of pretending that this representation was only involving the scalar partners of the 3-Gen MSSM. But it could also allow for this interpretation, as fermions with mirror fermions in the game. It could be argued that they were to be expected, as a GUT starting with real or pseudoreal representations -and this is why we do not usually see USP(2n) in the game- needs them.
Discussion here on obtaining SM fermions from unconventional representations of SU(5). It shows how the 15 branches to produce a left-handed quark SU(2) doublet. But my problem is, what about the right-handed quark singlet? It seems like it has to come from the 5 or 10. Does that mean that 15s and 24s are not enough?

Let me also expand a little on #252. The idea is that 15 x 15* = 1 + 24 + 200. So you could start with an N=1 U(5) gauge theory which only had 15 and 15* matter (chiral superfields), but the 24 would come out of the 15 x 15* meson superfield.

Also, a note for the future, I have discovered that there is an obscure ancient thesis containing a twistor model for quark-diquark systems. It is not online and the author (Aleks Popovich) left physics, but I hope we can track it down later this year.


Gold Member
Discussion here on obtaining SM fermions from unconventional representations of SU(5).
Well, not so unconventional, as the SU(3) subgroup there is still supposed to be colour, and not family. On the contrary, I was working with the expectation that both subgroups of SU(5) are "family groups", and colour is not seen until upgrading to SU(15).

Of course the SU(5) products can also be seen reflected in SU(15)

15 ⊗ 15* = (200) ⊕ (1) ⊕ (24) versus 15 ⊗ 15* = (224) ⊕ (1)
15* ⊗ 15* = (70′ ) ⊕ (50) ⊕ (105) versus 15* ⊗ 15* = (120) ⊕ (105)

but the amusing/careful point here is that while the 105 of SU(15) branches down to SU(5)xSU(3) producing colour triplets (15,3) plus colour sextets (10,6), this coloured 15 that we got is not the 15 we started from. The 15 of SU(15) also branches down, to (5,3).

On the other hand, the production of the 24 seems -at first glance, I have not checked in detail - to go similarly in both cases, as it is a colour singlet.

Also note that the need of working both with N and N* is the hint that really invites us to climb up to SO(30) or USp(30) at least.

The Physics Forums Way

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving