Kaluza-Klein theories in twelve dimensions and F-theory

[mitchell porter]

"Kaluza-Klein theories in twelve dimensions" ... and F-theory

This is a separate thread to discuss another idea by arivero, originally https://www.physicsforums.com/showthread.php?t=485247&page=4#63".

After all this LHC excitation, I am afraid i could go into hibernation for some period, but I want to say some words about this 313 GeV thing and how, to my regret, it could relate to extra dimensions. The point is that if we want quarks and leptons to stand in some symmetry group, the smaller candidate is SU(4), "Lepton number as the fourth color". The full group Pati Salam thing, SU(4)xSU(2)xS(2), is known to appear with 8 extra dimensions: it is the group of isometries of the manifold S5xS3, the product of the three-sphere with the five-sphere. It was argued by Bailin and Love that 8 extra dimensions are needed to get the charge assignmens of the standard model, but I am not sure if this manifold was used. Its role was stressed by Witten, who pointed out that the family of 7-dimensional manifolds that you get by quotienting this one via an U(1) action have the isometry group SU(3)xSU(2)xU(1).

I liked to think of this compactification as an infinitesimal extra dimension, partly because of the hint of F-theory, partly because thile the SU(4) group seems a need, I don't like to look at it as a local gauge group.

Again, this was well known lore of supergravity (and even in string theory) in the early eighties, but in the same way that the first revolution wiped gluons away, the second string revolution killed the research on realistic Kaluza Klein theories.

Some references:

http://www.sciencedirect.com/science/article/pii/0550321385902329" )

http://iopscience.iop.org/0034-4885/50/9/001". D Bailin and A Love, 1987. Rep. Prog. Phys. 50 1087.

The second article is a review of Kaluza-Klein theories and is noteworthy for going over the crucial arguments, due to Witten, about the difficulties of obtaining chiral fermions in KK theories, even if one adds torsion. Some of those arguments are from Witten's paper "Fermion Quantum Numbers in Kaluza-Klein Theory", which is apparently only available in the proceedings of the "Shelter Island II" conference from 1983.

It's curious that the first paper, on KK theories in 12 dimensions, is apparently completely uncited in the literature on F-theory, which suggests either that there's an obvious reason why one can't interpret F-theory in this way, or that the old KK lore really was forgotten.

 

[suprised]

It's curious that the first paper, on KK theories in 12 dimensions, is apparently completely uncited in the literature on F-theory, which suggests either that there's an obvious reason why one can't interpret F-theory in this way, or that the old KK lore really was forgotten.

Certainly "old" KK is not forgotten, but this paper is not relevant for F-theory. In F-theory the 11-th and 12th dimensions are on a different footing than the 10 others. I particular they live on an elliptic curve. So standard KK arguments do not apply here.

 

[mitchell porter]

Certainly "old" KK is not forgotten, but this paper is not relevant for F-theory. In F-theory the 11-th and 12th dimensions are on a different footing than the 10 others. I particular they live on an elliptic curve. So standard KK arguments do not apply here.

Yes, but one can still speak of compactifying eight dimensions in F-theory. So among other things, I'm interested in "conceptually interpolating" between "F-theory compactified on an elliptically fibered CY 4-fold", as considered by current research, and "twelve-dimensional Freund-Rubin compactifications", as considered by Bailin and Love - especially since both of them are aiming at physical realism. I think it would be instructive to see how far the analogy can be pushed, and where it breaks down.

 

[mitchell porter]

For example, this work from the 1980s can be used to construct Freund-Rubin compactifications of the form AdS4 x X8 or http://adsabs.harvard.edu/abs/1991NCimB.106..501Z".

So it would be interesting to look at degenerate versions of some of these 12-dimensional geometries, to see if they might correspond to the F-theory uplift of a 10-dimensional Type IIB AdS brane construction; and also to compare the effective theory arising from the brane construction, with the Kaluza-Klein theory arising from the non-degenerate, fully 12-dimensional geometry. There seems to be surprisingly little connecting F-theory with AdS/CFT so far (http://arxiv.org/abs/hep-th/9904050" )

 

[arivero]

Certainly "old" KK is not forgotten, but this paper is not relevant for F-theory. In F-theory the 11-th and 12th dimensions are on a different footing than the 10 others. I particular they live on an elliptic curve. So standard KK arguments do not apply here.

Yep. What is peculiar is that also the 12th dimension in KK is on a different footing; it is needed to get the correct charge assignments, but we really want the isometry group to live only in 11 dimensions, so it is was clear that some extra trick was needed here, perhaps to make it infinitesimal or to compact it separately. So F-theory comes as an inspiration because they need also some peculiar tricks to understand this extra dimension.

As fot the 11th dimension on a different foot, it is arguable... But please note that, due to the problem of chiral fermions, all dimensions above the 9th are "peculiar" for sure. The non chiral theory in nature is SU(3)xU(1), colour times electromagnetism. This is the isometry group of CP2 x S1, so it adds only five extra dimensions and the non chiral world lives in 9 dimensions. All the way from 9th to 11th to get the standard model group, and then from 11th to 12th to get the charges, and this phantom Pati-Salam (as ghostly that even Witten avoids to mention it when he produces S5xS3 in his paper on Kaluza Klein).

 

[suprised]

The essential difference between F-Theory and standard compactifications is that the former is non-perturbative and thus lies outside of the perturbative scope of the latter. So be careful with re-importing old concepts like KK. Actually the extra 2 dimensions in F-theory are ficticious, ie come from a formal trick, and it is unclear whether one should view it really as some (so-far unknown) 12dim theory. Probably not!

 

[arivero]

Via Clara (Nemo) I notice Atiyah's http://arxiv.org/abs/1108.5151 and very specifically sections 4.2, A.4 and A.5. The parallels between S4 and CP2 are revisited and expanded. Plus Hitchin metrics. Very interesting hints.