#### arivero

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The 54 down to [itex]SU(5) \times U_1(1)[/itex]

[tex]

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]

[tex]\begin{eqnarray*}

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)

\end{eqnarray*}

[/tex]

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.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,

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 site:arxiv.org flag, is very inneficient, so sure I am missing important references.

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