Minimum Fermion Content to Account for Standard Model: 12 of 8

[mormonator_rm]

The intuition is that the combination of two quarks into a QCD string is a boson, and the lowest energy state is an scalar (actually, pseudoscalar. But spin 0 anyway). When you use supersymmetry to uplift bosons to fermions, you need to match degrees of freedom. In four dimensions, it means that for each lepton or quark you need two spin 0 particles having the same charge.

I am a bit unsure about how it works in dimension greater than 4, say 10 or 26. For instance in dimension 10... do they match one fermion against eight spin 0 particles? The source of my troubles with higher dimensions is that you seem to need extra symmetries beyond CPT if you want to put some order in the components of a solution of Dirac equation.

Okay. I get it now. I'm sorry, I've been a little out of it lately.

Well, 10 or 26 dimensions should, indeed, be problematic. I would imagine. So, now I see what your getting at, and I see the problem too. Good luck, I can't help you a whole lot because I'm not that keen on String Theory. In all honesty, I tend to try to avoid String Theory because of the background dependence, and because I haven't bothered to study the latest on it in a few years, also because of the background dependence. Basically, I don't like background dependence...

 

[arivero]

In all honesty, I tend to try to avoid String Theory because of the background dependence, and because I haven't bothered to study the latest on it in a few years, also because of the background dependence. Basically, I don't like background dependence...

Yep I have also tryed for years to avoid string theory... and now this thing has imposed itself on my shoulders

 

[jal]

http://arxiv.org/abs/0804.0637v1
A Complete Classification of Ternary Self-Dual Codes of Length 24
Authors: Masaaki Harada, Akihiro Munemasa
(Submitted on 4 Apr 2008)
----------
http://arxiv.org/abs/0805.2205
Mass formula for self-orthogonal codes over Z_{p^2}
Authors: Rowena A. L. Betty, Akihiro Munemasa
(Submitted on 15 May 2008)
A quaternary code is said to be even if the Euclidean weight of every codeword
is divisible by 8. Every quaternary even code is self-orthogonal.

 

[Gokul43201]

What is the relevance of the above papers to this thread? I don't see any.

 

[arivero]

What is the relevance of the above papers to this thread? I don't see any.

I see a far far far one; but perhaps jal would like to explain. Of course, from time to time it is suspected that the 24 in string theory (25-1) and the 24 in lattices have a common origin.

 

[jal]

I doubt very much that my comment could help you.
Rather, you comment would help me.

The title, “Who has 12 of 8?”, seems to solicit an answer that would be based on some of the quantum gravity approaches, which of course is based on how the universe made symmetry and made phase changes. Those two papers seem to cover just about all of the possibilities of making symmetries that could be made.
It appears that the initial symmetry of the universe could have been the Leech Lattice.
The symmetries of confinement have been reduced to within “a hard shell” or “dust bag” and it does not require The Leech Lattice as an explanation.
jal

 

[arivero]

Leech Lattice is the Holy Graal of Conway's team. There is some gauss thing, $$e^{\gamma/8}$$, for unimodular selfdual whatevers. And of course there is the famous diofantine
$$\sum_{l=1}^n l^2=m^2$$.
But one needs a very high level of sphere packing gimnastics to understand the connections; I am afraid I am not trained on it. Someone?

 

[arivero]

12 of 8 = 6 of 16

Perhaps it could be more interesting to consider 6 groups of 16 particles:

-The groups of 6 appear in some perspectives of Lisi gadget

-At all, the fermion cube can be considered as having 16 degrees of freedom, if it includes the spin of each spinor (but not the antiparticle, then it should be 32).

-N=4 SUSY has a 16-plet. And...

-SO(32) heterotic string theory, when compactified down to 4 dimensions via a 6-torus, groups 6 of these 16-plets, and thus 96 scalar fields. Or perhaps the 16 is coming from a U(1)^16 related to the Cartan subalgebra of SO(32). Historically, this is the root of the first conjectured self-duality of string theory, back by Sen in hep-th/9402002.

 

[arivero]

12 of 8 = 48*2

-SO(32) heterotic string theory, when compactified down to 4 dimensions via a 6-torus, groups 6 of these 16-plets, and thus 96 scalar fields. Or perhaps the 16 is coming from a U(1)^16 related to the Cartan subalgebra of SO(32). Historically, this is the root of the first conjectured self-duality of string theory, back by Sen in hep-th/9402002.

Other cases:

http://arxiv.org/pdf/hep-th/9604164v2 gets groups of 96 scalars in compactification of M theory to 2 dim, via K3xK3xS1.

hep-th/9508058 "the heterotic string on a Narain torus at a generic point of moduli space is obtained by compactifying the ten-dimensional N = 1 Maxwell/Einstein supergravity theory on a six-torus" and then one gets 96 scalar field there, same as Sen.

hep-th/0411118 gets 48 out of SU(D+1), D=6, and/or " and 48 spinorial 16 of SO(10),
sixteen from each sector", or hep-th/0401114 "NAHE set models have N = 1 spacetime supersymmetry and 48 chiral generarions"

http://arxiv.org/pdf/hep-th/0409132 http://arxiv.org/pdf/hep-th/0501041 hep-ph/9604302 etc (NAHE sets) "projects the above N = 2 spectrum to an N = 1 gravity multiplet, the dilaton chiral multiplet, and 6 untwisted + 48 twisted geometrical chiral multiplets."

Froggart and Nielsen reminder us that "Similarly, including three right-handed neutrinos, the biggest anomaly free subgroup of U(48) is SO(10)". Chou and Wu have tried to produce families from a discrete \Delta 48 family group upon SO(10).

hep-th/9603081v2 describe D-particles with "48 fermionic ones"

d=5, N =8 supergravity is described in a number of papers and it has 48 "symplectic?" Weyl fermions. Eg hep-th/0205235v1 hep-th/9812092. I am kind of amused because I think that Kiritsis book list this content, or a very similar one, for d=4 compactifications.

 

[arivero]

d=5, N =8 supergravity is described in a number of papers and it has 48 "symplectic?" Weyl fermions. Eg hep-th/0205235v1 hep-th/9812092. I am kind of amused because I think that Kiritsis book list this content, or a very similar one, for d=4 compactifications.

Let me confirm that Kiritsis refers to d=4. At the end of section 9.1, page 223, he says that

"describe the toroidal compactification of type II string theory to four dimensions... We compactify in a six torus... The two Majorana-Wey gravitini and fermions give rise to eight D=4 Majorana gravitiny and 48 Spin 1/2 Majorana fermions"

.

I really do not catch it. Does string theory predict the fermion content of the standard model? Then, what are we discussing about?

About NAHE sets, the original preprints of Antoniadis et al circa 1989 are available in KEK. It is sort of intriguing because no connection is done initially to the general statement that relates the number of generations to the euler characteristic of the compactified manifold. It is only later that Faraggi (and see hep-th/0403272) connects them to Z2xZ2 orbifolds and that some papers in the arxiv (eg http://arxiv.org/abs/hep-th/0403058 ) mention the Euler Characteristic of NAHE based models. It is amusing that the NAHE set starts with 48 generations, and then the number goes down to 3 via the orbifolding and Wilson lines. It is unclear to me if the NAHE conditions have already been reformulated as Calabi-Yau, or proved different.

In this context (Faraggi research), Lubos likes to mention hep-ph/9405357 and its prediction of top mass neglecting some near predictions of Ibañez and Alvarez-Gaume teams. In fact, the attempt of prediction via NAHE is already in the original papers.

 

[arivero]

Hmm perhaps the point is that the 48 fermions are Majorana while the SM ones are Weyl. Or perhaps it is that string theoretists are lazy about model building, and they conceded NAHE as being the definitive model, not researching more phenomenology for 30 years.

 

[arivero]

A different view on this: to consider 3x(4x8)=4x24 insread of 12x8,

here "3" is the generation number. So there are two arrangements for the 8-plets:

I - labeled via SU(3)xU(1)
II - labeled via CPT.

In the case I, the uncoloured 8-plet contains the electron and the neutrino, and the coloured ones have a pair of quarks u,d

In the case II, the whole "fermion cube" is fit into the 8-plet, and the four 8-plets are labeled via the operations of parity and charge conjugation.

I wonder if there is some duality between I and II.

of course, when we have 3 generations there are an expansion to four 24-plets.

It is funny, but one is left thinking that this string people were in the right path twenty years ago.