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## Higgs, SuperStrings and Kaluza Klein

• 13D "S-Theory" M4 x S9 has as isometry group SO(10) for the internal S9 space.
• 12D "F-Theory" M4 x S5 x S3 has SO(6)xSO(4), locally SU(4)xSU(2)xSU(2)
• 11D "M-Theory" M4 x ((S5xS3)/S1) has isometry group SU(3)xSU(2)xU(1)
• 10D is target space of SuperStrings, and perhaps also of Connes Chamseddine Marcolli
• 9D "T Duality Limit" M4 x CP2 x S1 has isometry group SU(3)xU(1)

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 Blog Entries: 6 Recognitions: Gold Member I have purposely waited some views to allow you some personal meditation of these facts. If not done, please take your time now. My view is that there are two important jumps here which are related to the standard model: that from 12D to 11D, and that from 11D to 9D. From 12D to 11D, string theoretists usually speak of "two times", "infinitesimal size of the 12th dimension", or similar things. Connes asks for a maximal subalgebra of $C+H+H+M_3(C)$, with a non trivial dirac operator. Mohapatra, Pati and all the guys noticed that this non trivial quotient was related to the special role of neutrinos. And Witten noticed that it was a $S^1$, or $U(1)$ non trivial action on the manifold of extra dimensions, and got a family of spaces in 11D having the symmetry of the standard model. So everything is about a very special breaking, not exactly your traditional spontaneous one, that selects the special role of right neutrinos and gives them a big majorana mass (surely inverse of the "infinitesimal" dimension) 11D to 9D is in my opinion related to the "network of dualities" that involved the T dualities and the discovery of M-theory itself as a limit up from the ten dimensional superstring. Here we are seeing the SU(2)xU(1) breaking of the standard model; the size of these 10th and 12th dimension, plus the scale of the string tension, are surely enough to imply all the parameters of our beloved electroweak symmetry breaking: the W and Z, and the higgses (if more than one) and the "electroweak vacuum". Last in 9D, we are left with the unbroken groups having only non-chiral gauge symmetries: QCD and Electromagnetism.
 Blog Entries: 6 Recognitions: Gold Member Let me note that Motl has started a post near to this subject http://motls.blogspot.com.es/2012/08...tries-are.html which will surely be more interesting that this thread. Still, you are invited to comment here too :-D

## Higgs, SuperStrings and Kaluza Klein

Yours is not present in wikipedia, so, it is more interesting. I am waiting for more stuff, like:

The dynamics of a 4d manifold, "a pure gravitational one", which is equivalent to a charged 4d object living in a 13D space.

 Quote by arivero Let me note that Motl has started a post near to this subject http://motls.blogspot.com.es/2012/08...tries-are.html which will surely be more interesting that this thread. Still, you are invited to comment here too :-D

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 Quote by MTd2 Yours is not present in wikipedia, so, it is more interesting..
Well, it is not in wikipedia because practicioners do not take time to update it, but I was not planning to have a thread very speculative.

 Quote by arivero but I was not planning to have a thread very speculative.
So, what do you want with this thread? It is not like you are asking to solve any text book problem....

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 Quote by MTd2 So, what do you want with this thread?
Awareness.

It doesn't matter (to me; mods can have different opinion ) if it evolves toward textbook exersices or towards speculation. Did you know that only 7 extra dimensions, ie D=11, are enough to produce the Standard Model gauge group via Kaluza Klein? Even Lubos started its own post denying it:
 The Standard Model gauge group SU(3)×SU(2)×U(1) has the rank (the maximum number of independent yet mutually commuting generators) equal to 2+1+1=4. Because you need at least a pair of dimension for each rotation in the rank, you would need at least 8 compactified dimensions to geometrize the Standard Model gauge group as the isometry of some Kaluza-Klein dimensions. Well, the actual minimum number of dimensions is even higher because the non-Abelian group is much larger than its commuting generators.
 So, say something about my random idea!

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 Quote by MTd2 So, say something about my random idea!
OK, in your case the question, whose answer I ignore, is, does http://arxiv.org/abs/hep-th/9607112 S-Theory admit a compactification on the 9-sphere, so that the Kaluza Klein group is SO(10)?

And then you could follow asking yourself: Which are the charges of the low energy states in such compactification, if it exists? How could either SU(5) or SU(4)xSU(2)xSU(2) relate to it? Is there some network of dualities between M-Theory, F-Theory and S-Theory which could, at the end, be related to this SO(10) group? And so on.

I confess I have never been interested on S-Theory, so I have no idea if it is really important to understand the rest of the structures. I mention it now only to obey your command :-D
 It seems that every dimension has some sort of symmetry of a larger and larger GUT. Should we see E6 somewhere?
 Wik says E6 is the isometry group of the "bioctonionic projective plane".

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 Quote by mitchell porter Wik says E6 is the isometry group of the "bioctonionic projective plane".
Yep, it seems that the bosonic string theory has some say with the exceptional groups. E6/F4 is itself of dim 26; the bioctonionic proj plane is dim 32, but F4 are isometries of its kin "octonionic proj plane", only of dim 16, surely related also to the compactification lattice that interplays between 26 and 10. .. again, we see that the " stringy enhanced symmetries are natural, important, and cool", but that is really my point, that the coolness of such symetries has distracted the crowd from the real fundamental issue, which is dim 10, until the point that people knows of these symmetries while they ignore the ones at the start of the thread.
 Blog Entries: 6 Recognitions: Gold Member Now, consider the standard model at high and low energy? Which gauge theory approximates the low energy standard model? Which gauge theory approximates the high energy standard model?
 Blog Entries: 6 Recognitions: Gold Member And for the Higgs, my question could be, what mechanism does string theory offer to produce massive gauge bosons? Polchinski in his textbook tell us that the self-duality point can be interpreted Higgs-like, that SU(2)xSU(2) becomes U(1)xU(1) outside of this point and then the other four bosons become massive. Similar themes I have read about the squashed spheres in Kaluza Klein. I would expect some similar mechanism also in lens spaces, but I haven't found any paper trying it.
 Blog Entries: 6 Recognitions: Gold Member And for the Higgs, my question could be, what mechanism does string theory offer to produce massive gauge bosons? Polchinski in his textbook tell us that the self-duality point can be interpreted Higgs-like, that SU(2)xSU(2) becomes U(1)xU(1) outside of this point and then the other four bosons become massive. Similar themes I have read about the squashed spheres in Kaluza Klein. I would expect some similar mechanism also in lens spaces, but I haven't found any paper trying it.

 Quote by arivero 13D "S-Theory" M4 x S9 has as isometry group SO(10) for the internal S9 space. 12D "F-Theory" M4 x S5 x S3 has SO(6)xSO(4), locally SU(4)xSU(2)xSU(2) 11D "M-Theory" M4 x ((S5xS3)/S1) has isometry group SU(3)xSU(2)xU(1) 10D is target space of SuperStrings, and perhaps also of Connes Chamseddine Marcolli 9D "T Duality Limit" M4 x CP2 x S1 has isometry group SU(3)xU(1)
The list of theories you mention, and their corresponding dimensions, may be tied to gradings of the exceptional Lie algebras (and lifted to their exceptional groups). I've discussed this with I. Bars and L. Boya before. The patterns are most visible with the real forms of the exceptional Lie algebras. Consider the following gradings:

g = E6(-26) , g(0) = so(1,9) + R , g(-1) = M1,2(O).

g = E7(-25), g(0) = so(2,10) + R, dimR g(-1) = 32, dimR g(-2) = 1

g = E8(-24), g(0) = so(3,11) + R, dimR g(-1) = 64, dimR g(-2) = 14.

In these examples, one can see the g(0)'s contain 'rotational' parts for (would be) space-times of signatures (1,9), (2,10), and (3,11) respectively, along with an extra 'translational' component. In the E7(-25) and E8(-24) gradings, the g(-1) part gives the corresponding spinor dimensionality.

Note: For E6(-26), the spinor dimension is more easily seen in the g(-1) part of the 5-grading: g = E6(-26), g(0) = so(8) + R + R, dimR g(-1) = 16, dimR g(-2) = 8.

By dimensionality of space-times and spinors, one can create the following maps:

10D superstring theory, 11D M-theory → E6
12D F-theory, 13D S-theory → E7
14D Unknown (3,11) theory "T-theory" and its 15D completion "U-theory" → E8

In discussing a three-time theory with I. Bars, he admitted it could be a possibility if one can eliminate all ghost states that arise, which he managed to do with S-theory.

One can also look to the other real forms of the exceptional algebras and their gradings, which include other possible signatures:

g = E6(6), g(0) = so(5,5) + R , g(-1) = M1,2(O')

g = E7(7), g(0) = so(6,6) + R, dimR g(-1) = 32, dimR g(-2) = 1

g = E8(8), g(0) = so(7,7) + R, dimR g(-1) = 64, dimR g(-2) = 14.

Here, the time-dimensions have increased, while the spinor dimensions remain the same. String/M-theory in signatures with more than one time dimension have been discussed by C. M. Hull (hep-th/9807127).

Going to the fully complexified algebras, one has the gradings:

g = E6C, g(0) = so(10)C + C, g(-1) = M1,2(O)C

g = E7C, g(0) = so(12,C) + C, dimC g(-1) = 32, dimC g(-2) = 1

g = E8(C), g(0) = so(14,C) + C, dimC g(-1) = 64, dimC g(-2) = 14

where the space and time components are unified and the spinor dimensions are complex. In Hull's paper, he refers to a complex string/M-theory, stating that the "new theories are different real forms of the complexification of the original M-theory and type II string theories, perhaps suggesting an underlying complex nature of spacetime."

There exist other gradings, but the ones mentioned are the most suggestive in looking for hints of string, M, F and S theory inside such exceptional structures. Moreover, such gradings may also hint at new 14D and 15D theories that are yet to be found. Finding theories inside exceptional structures is very much like Lisi's approach and I have also discussed these gradings with him on several occasions. It is also worth mentioning the real forms of the exceptional groups arise as U-duality groups in (toroidally compactified) M-theory and extended supergravity and act on the charge space of extremal black holes.

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 Quote by kneemo The list of theories you mention, and their corresponding dimensions, may be tied to gradings of the exceptional Lie algebras (and lifted to their exceptional groups).
Hmm but this is not different from these groups appearing when you do a compactification to D=3 or D=2, is it? There were dubbed "non compact global symmetries of compactified gravity"

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