Why is the SU(3)xU(1) Group Essential for Dirac Fermions?

[arivero]

This is a companion question to https://www.physicsforums.com/threads/why-su-3-xsu-2-xu-1.884004/

Of course the Higgs mechanism over the standard model produces this low-energy group, SU(3)xU(1), which acts on Dirac fermions (this is, no Left-Right asymmetry anymore).

Is there some reason, beyond experimental observation, to need this group particularly, and the precise way it acts? Given SU(3)xSU(2)xU(1), are we already forced to choose a Higgs mechanism that hides the chiral (axial?) part of the electroweak force?

 

[arivero]

By the way, the fact of SU(3)xU(1) acting on Dirac fermions has the interesting consequence of bypassing the usual objection against Kaluza-Klein theories; which are discarded because it is not possible to put chiral fermions on (most of) then. Here in principle we could use a 9-dimensional space, one dimension less than string theory, with compactification manifold CP2 x S1 (or CP2 x CP1 if you prefer).

 

[mitchell porter]

The Witten manifold for the SM gauge group, M111, has a quotient with symmetry group SU(3) x U(1)^2... see page 5 here.

As I understand it, the quotient applies to a U(1) factor within SU(2), i.e. there is a one-parameter set of "rotations" of the manifold onto itself, and to form the Z_k quotient, you divide that circle of rotations into k segments, and then only keep enough of M111 that would correspond to one "segment". Like replacing a pie with just one slice of the pie, and then folding the slice over to make a cone shape.

If you do that, all that is left of SU(2) is a different U(1) subgroup. Meanwhile, M111's original U(1) is untouched, so the remaining symmetry of this "M111/Z_k" manifold (which is still 11-dimensional) is SU(3) x U(1) x U(1).

So I'm wondering if one could pursue your program of a d=9 Kaluza-Klein model for QCD+QED on "M111/Z_k" with two compactification scales. In d=4, you have QED+QCD; in d=9, you have "Kaluza-Klein QED+QCD"; and in d=11, you have "something like" the full SM gauge group.

One might go further and guess that the transition from d=11 to d=9 is associated with supersymmetry breaking, and the transition from d=9 to d=4 with electroweak symmetry breaking. For the first transition, I might seek inspiration in the neglected case of G2-MSSM with few moduli and high susy scale (see page 7, "reason b", here). For the second transition, I might look to "postmodern technicolor", in which chiral symmetry breaking of technicolor, contributes to electroweak symmetry breaking.

 

[arivero]

One might go further and guess that the transition from d=11 to d=9 is associated with supersymmetry breaking, and the transition from d=9 to d=4 with electroweak symmetry breaking.

Hmm? I almost certainly expected it to be the other way: d=11 goes to d=9 because of electroweak symmetry breaking; or even including LR-breaking if you are considering M111 (which is SU(3)xSU(2)xSU(2)). Then something causes d=9 down to d=4 but I can not guess what it is; susy breaking could do find here.

Also, note that we could go d=12 to d=9 if we consider we are starting, as Witten did, from S3xS5, with both Pati-Salam and L-R symmetry.