if you build a Kaluza-Klein type theory, but use the SM gauge group U(1)×SU(2)×SU(3) instead of U(1), what do you get?
If you want to use U(1)xSU(2)xSU(3), you get gravity over a 11 dimensional manifold, such the extra seven dimensional manifold is of the kind produced by quotienting $S^3 \times S^5$ by an orbit of U(1). A particular space in this family is $S^3\times CP^2$, you can learn that the group of isometries of $CP^2$ is SU(3) and the group of isometries of $S^3$ is obviously SO(4), so SU(2)xSU(2). More general spaces of this kind can be obtained by using a generic lens space instad of $S^3$; remember that lens spaces interpolate between $S^3$ as fiberes product of $S^1$ and $S^2$ and the plain product $S^2 \times S^1$. (This is already a bit beyond the answer, but I mention it becasue my first question in Physics.SE was about if this interpolation was a kind of Weinberg angle).
The dimension of a Lie Group is equal to its number of generators, so G=U(1)xSU(2)xSU(3) has, as a manifold, dimension 1+3+8=12. Such manifold has an action with GxG, which is overkill. So we can quotient the manifold using a maximal non trivial subgroup of G, in this case H=U(1)xU(1)xSU(2), and use instead the manifold G/H. Thus the number of dimensions that we need is $1+3+8-(1+1+3)=7$.
The ways to map H into G are not unique, and in the particular case of the SM group this creates a 3-parameter family of manifolds, and each of them seems to have, according Salam et al, a 2-parameter family of metrics. In some special cases of this parameter space, as the aforementioned $S^3\times CP^2$, some extra symmetries can appear.
I am not sure, but it seems that before Witten the technique to put "$G$ instead of U(1)" was really to put the whole Lie manifold as compact space, and then act on it with $G\times G$. A particularly intriguing case is when $G$ has the topology of an sphere, and then the maximal possible number of isometries. So $S^1$ and $S^3$ had naturally attracted some attention, and Adam theorem could have pushed some interest on $S^7$.
But even if this (Kaluza-Klein)+(SM gauge) theory only describes interactions, isn't that okay, or at least a big help?
It seems that it does not help, and I am as surprised as you.
The question of fermions "by hand" goes beyond the chirality problem. It was a program, led mainly by Salam, that an analysis of the compactification manifold and its tangent plane should reveal the charge assignments. For the SM-like manifold in 7 dimensions, the program fails; you can not find the charge assigments that the standard model has. It was noticed later, by Bailin and Love, that by going to 8 extra dimensions the problem could be solved, but further research was not pursued.
A reasonable inquiry is how the jump to 8 and eventually 9 dimensions relates to Pati-Salam, SU(5) and SO(10). Of course SO(10) needs nine extra dimensions (It is the isometries of $S^9$), and the projections down to the standard model seem very much as recent work of John Huerta. Other interesting question, to me, is if the extra dimension, from 7 to 8, can really be a local gauge symmetry, given that we have reasons to keep ourselves in D=11 at most. When one notices that the extra dimension is the origin of $B-L$ charge, that is interesting.
History
You can also check SPIRES for the history of Witten involvement in Kaluza Klein: FIND A WITTEN AND K KALUZA-KLEIN (edit:link changed towards inspire)
He has four papers with the Keyword "Kaluza Klein". The first of them is "Realistic Kaluza Klein Theories". It is the start of the KK trend, not the end. All the relevant papers come because of it. Do an
FIND K KALUZA-KLEIN AND TOPCITE 50+ AND DATE BEFORE 1990 AND DATE AFTER 1975
And order by increasing date. You will notice the works of Salam et al, Pope, Duff, all of it. The difference with the previous, and later, research, is that in this timelapse KK was considered seriously as in the original proposal, while generic references about KK in modern literature are really about compactification from higher dimensions; in some cases the fields comming from KK are even a nuissance to avoid.
I do not know who invented the late excuse that "Realistic Kaluza Klein" killed the research on KK; it appears very frequently in folk introductions to compactifications in string theory. More rarely, some person notes the contradiction and quotes instead the last paper of Witten on the topic, Shelter Island II, which has a more deeper discussion on the chirality problem, and even hints -or I read between lines- the question of singularity or regularity of the manifold, so that the late proposed solution to the fermion problem (see Moshe answer) is not so surprisingly ironical, really it was there from the start.
The topic of Kaluza Klein, or more properly of using the gauge fields of Kaluza Klein as physical fields, was abandoned in 1984 with the second superstring revolution. Ten dimensions were more interesting that eleven, and then you have not enough room to produce the SM group in a pure way from KK, so why bother? A mix of crowdthinking with "publish or perish" led to the end of the research, as most of the easy topics on KK had been covered in the interval (from 1981 to 1984), and some others were common to any extra dimensional theory: compactification, stability, etc... Nobody was even worried by the estability, because at that time it was believed that an AdS spacetime was reasonable, and then some compactification mechanisms from $M^{11}$ to $AdS \times M^7$ were known. An important role in this mechanism was the 84-component tensor that is added to the 44-component graviton in supergravity; some years later it should be recognised as the starting point of M-theory.
This post has been migrated from (A51.SE)
Q&A (4871)
