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While QCD is a typical product of AdS/CFT and some other research trends in extra dimensions, I have never found in the literature an example producing the non-chiral part of the standard model, colour plus electromagnetism, or even color alone, from D=9 Kaluza Klein.

In principle such theory could be obtained

  • by compactification on the 5-sphere $S^5$ and adding some Higgs to break $SO(6)$, or

  • by compactification on the product manifold $CP^2 \times S^1$, producing directly the gauge group $SU(3) \times U(1)$.

Besides, QCD alone could be got from a $D=8$ theory on $CP^2$.

The traditional argument about the absence of chiral fermions do not apply here as both QCD and EM are defined with Dirac fermions. So if there is a non-go theorem at work, forbidding the scheme, must be a different one. Here the question: Is there one? Or, as a counter-proof of my question, is the example actually done in the literature and it only happens that I have not searched deep enough?

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  • $\begingroup$ A collateral talk point is if this approach connects or not with the well worked D=10 $AdS^5 \times S^5$ $\endgroup$ – arivero Jul 31 '15 at 11:03
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While it is certainly possible to get an $SU(3) \times U(1)$ gauge group from the metric alone, if one started with a 9d theory, there are several issues with using Graviphotons as gauge bosons in a 4d theory.

Most prominently, in addition to vector bosons you will always create scalars in the adjoint from internal components of the metric, which we do not observe.

Furthermore, while the effective gauge group active at everyday energies is indeed a vector-like $SU(3)_\mathrm{c} \times U(1)_\mathrm{em}$, we do know for certain that there are effects that are due to a chiral $SU(2)_L \times U(1)_Y$ which is broken to $U(1)_\mathrm{em}$ via the Higgs mechanism. I'm not aware of someone reproducing spontaneous symmetry breaking on off-diagonal blocks of a metric (I think Witten tried some time in the 80s, but I don't have there reference at hand right now).

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  • $\begingroup$ The last part of your answer should be indeed a topic for a different question... some D=11 to D=9 reduction emulating SSB $\endgroup$ – arivero Aug 5 '15 at 13:15
  • $\begingroup$ Well, the part of the answer about the existence of new scalars is relevant enough to accept the answer. Still, this is a generic feature of all Kaluza Klein theories, isn't it? Without an analysis of this sector and their masses, to suggest it is not realistic is mostly a guess. $\endgroup$ – arivero Aug 7 '15 at 21:52
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    $\begingroup$ @arivero It is notoriousely hard to make these scalars massive. They appear in the metric, so the allowed couplings are highly restricted. Most importantly, you can not write down a tree-level potential for them, but you need to employ some mechanism of moduli stabilization. $\endgroup$ – Neuneck Aug 8 '15 at 6:45
  • $\begingroup$ Btw, @neuneck , perhaps you can be interested also in my question in MathOverflow, mathoverflow.net/q/213001/4037 Conditions for underlying space of an orbifold Tn/Γ to be a sphere? $\endgroup$ – arivero Aug 9 '15 at 0:01
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With an answer selected (and bounty awarded) it is time to open a community wiki for explicit references on work along the line of getting QCD + EM, or alternatively QCD alone or QCD + "4th colour" extracting the group from the extra dimensions.

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