# Is there some no-go theorem for $D=9$ Kaluza Klein QCD+EM?

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?

• A collateral talk point is if this approach connects or not with the well worked D=10 $AdS^5 \times S^5$ – arivero Jul 31 '15 at 11:03

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.
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).