# How many extra dimensions?

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## Main Question or Discussion Point

SU(3) needs to act on a minumum of 4 compact dimensions.
SU(2) needs a minimum of 2 dimensions
U(1) needs a minumum of 1 dimension

Thus SU(3)xSU(2)xU(1) needs 7 extra dimensions. Because of this, string theoretists do not use the extra dimensions to generate the gauge groups, and they use instead -wrongly, IMO- the flavour group of the superstring, because in string theory it is a local gauge group too.

Note that if SU(2) were wiped out, we would need only 5 extra dimensions.

Now, the standard model is not SU(3)xSU(2)xU(1), not SU(3)xU(1), but something between: the electroweak symmetry is broken but not deleted.

Question is: does the breaking of the group make room to use a dimension lower than seven? In this view, in the limit were the Higgs goes to zero symmetry is restored and the dimension should be seven (or 11, if you wish to count spacetime). But with the Higgs around, we could be using a dimension less than seven.

Intriguingly, Connes and Chamseddine concluded, in the 2006, that the dimension of the finite part of the standard model spectral triple was 6 (mod 8). And the CC model is forced to incorporate the Higgs automagically.

## Answers and Replies

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Let me expand a bit further, hoping to have some answer. Let G be SU(2), and allow it to act on a 2 dimensional sphere. All the three generators W0, W+, W- are well defined there. But now consider SU(2) acting on a 1 dimensional circle. The action of W+ and W- becomes undefined. For all the accounts, the SU(2) symmetry has broken to U(1).

Is there some way to use this kind of argument to reproduce the symmetry breaking of the ElectroWeak GSW model?

Hmm, I am confused. Could you please help me understand? I had been under the impression that string theory requires extra dimensions because useful theory requires the Conformal anomaly/Weyl anomaly to cancel; and that the lowest number of dimensions in which this happens is 26 for the normal string, and 10 for the superstring-- as far as I know this is the case for all string theories, REGARDLESS of what actual particle fields exist in that theory. What do you refer to when you talk about SU(3)xSU(2)xU(1) needing to "act on" 7 extra dimensions?

Thanks!

In the ElectroWeak GSW model the relationship of two fundementals are combined, leaving the others to fend for themselves.

In the 1 dimensional circle it's relationship with the whole universe is unified.

Will it not be difficult (or impossible..) to reconcile the two concepts?

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Intriguingly, Connes and Chamseddine concluded, in the 2006, that the dimension of the finite part of the standard model spectral triple was 6 (mod 8). And the CC model is forced to incorporate the Higgs automagically.
Connes mentions this interesting/intriguing "coincidence" in an interview given to our french/german TV channel (not seen mentionned anywhere else). I guess he did investigate this.

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Hmm, I am confused. Could you please help me understand? I had been under the impression that string theory requires extra dimensions because useful theory requires the Conformal anomaly/Weyl anomaly to cancel; and that the lowest number of dimensions in which this happens is 26 for the normal string, and 10 for the superstring-- as far as I know this is the case for all string theories, REGARDLESS of what actual particle fields exist in that theory. What do you refer to when you talk about SU(3)xSU(2)xU(1) needing to "act on" 7 extra dimensions?
I refer to Kaluza-Klein theories. As described in Witten's article "Search for a realistic Kaluza Klein Theory".

Contrary to popular knowledge, superstring theory does not get the gauge fields of the standard model from extra dimensions. It was asserted as impossible because of the argument I sketched: that you need a minimum of seven extra dimensions.

In the aforementionated article, Witten gives as a merit of supergravity to need exactly seven dimensions at most, so the maximal of supergravity intersects with the minimum of KK theories. Supergravity was discarded, but years later it was discovered that a limit of superstring theory happens to have 11 dimensions, thus seven extra dimensions. So in this limit it should be possible to get the standard model gauge group out from the extra dimensions, instead of the usual trick of extracting it as a subgroup of the flavour group.

That is very interesting, thank you.

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Connes mentions this interesting/intriguing "coincidence" in an interview given to our french/german TV channel (not seen mentionned anywhere else). I guess he did investigate this.
Yes and no; he is not a big fan of string theory, as you probably can to infer. Only a couple of collaborators are interested in strings: Lizzi and, occasionally, Chamseddine. When we discussed of it back in 2006, nobody was able to extract a hint of this coincidence. Now, if it were possible to interpret the Higgs field as an interpolation between six and seven extra dimensions, THAT should be a clue.

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In the ElectroWeak GSW model the relationship of two fundementals are combined, leaving the others to fend for themselves.

In the 1 dimensional circle it's relationship with the whole universe is unified.

Will it not be difficult (or impossible..) to reconcile the two concepts?
One though more... SU(2)xU(1) needs three dimensions (S2 times S1 if you wish); U(1) needs one dimension (S1, the circle). The fall down from SU(2)xU(1) to U(1) would be the correspondence with this jump from three down to one. But it is a *complete* breaking; in this setup the masses of W and Z should be infinite, so that only an U(1) extants, for the photon.

Question is, if the case with W and Z massless lives on dimension three, and the case with W and Z infinitely massive lives in the 1 dimensional circle... what about the intermediate case? Yes it seems difficult to reconcile the two concepts. But it seems worthwhile.

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Yes it does seem worthwhile I commend your intellect.

Basically W and Z in both cases whether infinite or massless, ultimately can have an attibutable value of zero as they equally obliterate anything meaningful when used as a multiplier.

I'm not sure how to extrapolate the intermediary from the two extremes.

Or we take massless to be 1 and infinity to be 0, or vice versa.

In an abstracted single dimensional view of our universe I tend to describe the smallest indivisible element of space time as 1 and the smallest indivisible element of mass to 0.

As a result in that model I quantify the amount of mass present in space time, as a displacement of space time.. as I ascribe no value to mass and view it as a void.

I wonder if there is any way to apply that style of reversal of perspective on this problem?

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Two notes about continuations of this thread:
- it relates to the PF thread
https://www.physicsforums.com/showthread.php?t=224509
- some expanded comments appear at lubos blog:
https://www.blogger.com/comment.g?blogID=8666091&postID=8616095981909796543&pli=1

arivero said...
What I would like to see, someday, is a model where all of the Standard Model gauge bosons come from Kaluza Klein bosons.
Why? Well, in the limit of infinitely massive W and Z, the symmetry of the standard model is SU(3)xU(1), so a Kaluza Klein with 5 extra dimensions. In the limit of massless W and Z, the symmetry is SU(3)xSU(2)xU(1), so a Kaluza Klein with 7 extra dimensions. I would expect that all of these duality tricks jumping between 9, 10 and 11 dimensions will correspond to the phenomena we already detect in the standard model.

Jun 7, 2008 12:10:00 AM
Blogger Lumo said...
Dear arivero, there's a theorem that was proven soon after type II string theory started to be considered seriously for phenomenology that implies that you can't get the SM gauge group in the purely KK fashion, as an isometry of a hidden manifold.

While I understand why you would view such a solution "elegant", it seems clear to me that a more refined look at the origin of gauge groups in string theory implies that yours is a naive belief.

There is nothing wrong for gauge groups to arise in the other, "non-classical" ways, including the built-in heterotic group and groups on D-branes and singularities. In some sense, all these pictures may be dual to each other so you can't say that one of them is worse than others.

Jun 7, 2008 5:46:00 AM
Blogger arivero said...
Dear Lubos,

If the theorem you refer to is simply the one by Witten on "realistic KK theories", then it is basically the result I state above: 9 dimensions for SU(3)xU(1), or 11 dimensions for the full SU(3)xSU(2)xU(1). If you are referring to other deeper theorem, please tell me!

My naive fascination here is that on one hand the standard model, with the finite, non zero, masses of W and Z, sits something in the middle of group theory, and string theory, with its web of compactifications, also happens to sit in a middle... whose extremal dimensions do coincide with the ones of the standard model.

Jun 7, 2008 9:15:00 PM

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Also, part of this discussion is VERY relevant to this thread

https://www.physicsforums.com/showthread.php?t=237296

It seems that the naivest descent from 11 to 9 dimensions is not the most realistic. It should start by taking the mass of Z to infinity, while W is still kept to zero. This amounts to break SU(3)xSU(2)xU(1) to SU(3)xSU(2)x1, which lives in 6 extra dimensions. Then the next step is to put mass of W to infinity, breaking further to SU(3)xU(1)x1, which acts naturally in five extra dimensions. Regretly in this approach one can not B and W3 to get Z and gamma as usual, because we got rid of B in the first step. Also, note that the naive descent is straighforward: no finite masses, just zeros and infinities.

nrqed
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I refer to Kaluza-Klein theories. As described in Witten's article "Search for a realistic Kaluza Klein Theory".

Contrary to popular knowledge, superstring theory does not get the gauge fields of the standard model from extra dimensions. It was asserted as impossible because of the argument I sketched: that you need a minimum of seven extra dimensions.
But what was the reason to dismiss seven extra dimensions? Am I correct that th point was that it was not possible then to have a chiral theory (because there would be ten dimensions of space and in that case one cannot have a chiral theory). Is my understand correct?

Contrary to popular knowledge, superstring theory does not get the gauge fields of the standard model from extra dimensions. It was asserted as impossible because of the argument I sketched: that you need a minimum of seven extra dimensions.
String theory is not subject to the same limitations to gain symmetry froms symmetry breaking that pointlike particles. The most straigforward models of compactifications on kaluza klein compatifications have a lot of simmetroy. Far enoguth to get the standard model. In fact the problem is that they have too much symmetry.

I must say that it surprise me your afirmation because i thought you had studied string theory. If not you could try to read a quick introduction in this entry (in spanish) of my blog http://freelance-quantum-gravity.blogspot.com/2007/10/nociones-bsicas-sobre-compactificacin.html of my blog about compactifications.

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Contrary to popular knowledge, superstring theory does not get the gauge fields of the standard model from extra dimensions. It was asserted as impossible because of the argument I sketched: that you need a minimum of seven extra dimensions.
I must say that it surprise me your afirmation because i thought you had studied string theory. If not you could try to read a quick introduction in this entry (in spanish) of my blog http://freelance-quantum-gravity.blogspot.com/2007/10/nociones-bsicas-sobre-compactificacin.html of my blog about compactifications.
You say that a 6 torus has the right dimensions... for what? It does not fit the standard model.

In most string models the gauge bosons come either from the symetry of the quantum string (E8xE8, SO(32) etc) or from D-Branes aka Chan-Paton factors. Thus from a quantum gauged "flavour" group, not from Kaluza Klein isomtries of extra dimensions. The role of compactification is a minor one. And the only link between extra dimensions and the gauge group is that 2^(D/2)=32

I am curious about orbifolds. Is there a rigorous definition of dimension of an orbifold? And of the group of isometries of an orbifold? I wonder if they can be used to fit the whole standard model as a group of isometries of a 6 manifold. And I would expect that we pay the price of spontaneus symmtry breaking of Z and W :-)

I am sorry I had done a longer reply, but I lost it :-( Basically, let me to tell that while my advisors and teacher are fond of string theory, I was not impressed and never looked at it in detail. It has been forced into me by empirical observations: Connes triple, my sBootstrap counting, the Z0 decay width...

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