# Higgs, SuperStrings and Kaluza Klein

1. Aug 2, 2012

### 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)

2. Aug 2, 2012

### arivero

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.

3. Aug 3, 2012

4. Aug 3, 2012

### MTd2

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.

5. Aug 4, 2012

### arivero

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.

6. Aug 4, 2012

### MTd2

So, what do you want with this thread? It is not like you are asking to solve any text book problem....

7. Aug 4, 2012

### arivero

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:

8. Aug 4, 2012

### MTd2

So, say something about my random idea!

9. Aug 4, 2012

### arivero

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

Last edited: Aug 4, 2012
10. Aug 5, 2012

### MTd2

It seems that every dimension has some sort of symmetry of a larger and larger GUT. Should we see E6 somewhere?

11. Aug 5, 2012

### mitchell porter

Wik says E6 is the isometry group of the "bioctonionic projective plane".

12. Aug 5, 2012

### arivero

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.

13. Aug 5, 2012

### arivero

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?

14. Aug 6, 2012

### arivero

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.

15. Aug 6, 2012

### arivero

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.

16. Aug 7, 2012

### 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). 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.

Last edited: Aug 7, 2012
17. Aug 7, 2012

### arivero

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"

18. Aug 7, 2012

### kneemo

What you mention is the way certain non-compact exceptional groups are seen from the perspective of M-theory on T^k (k-torus). But such non-compact groups only refer to the split real forms, e.g. E6(6), E7(7), E8(8). The other real forms such as E6(-26), E7(-25) and E8(-24) are not yet described in M-theory and are called the U-duality groups of the magic supergravity theories in D=5, D=4 and D=3, respectively. However, maybe there is a "dual" way to view the dimensionality, as the gradings of the exceptional Lie algebras suggests.

For example, consider the following grading of E8(-24):

g = E8(-24), g(0) = E7(-25) + R, dimR g(-1) = 56, dimR g(-2) = 1.

This grading is usually interpreted with the g(-1) part being the 56-dimensional Freudenthal triple system (black hole charge space in D=4), which is acted on by the E7(-25) (D=4 U-duality group). In the corresponding D=3 magic supergravity E8(-24) acts on the 56+1-dimensional "charge-entropy" space of the extremal black hole.

Yet, as mentioned before, there is also the 5-grading of E8(-24) with components:

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

For this one, I don't yet know the interpretation, but the grading is suggestive of a 14-dimensional theory with 64-component spinors.

The two different 5-gradings, are akin to two different "slicings" of E8(-24), giving rise to a "black hole frame" in the first 5-grading with dimR g(-1)=56 and the "spinor frame" in the second. Morphisms between these gradings would give rise to a type of "duality" between the frames which, individually, appear to describe different physical systems.

Last edited: Aug 7, 2012
19. Aug 7, 2012

### arivero

Yes I see. Reading the original papers there was a lot of excitation about all this, perhaps because it was the first time where they were able to show/use the exceptional structures. Still, I think that it is distraction; a lot of it must be tautological with all the other octonionic justifications of the existence of M-theory (the brane scan, etc) and part of it could be just the KK version of a catalogue of subgroup branchings and breakings as we reduce dimensions. Some of it could, in the long rung, explain the pieces of the SM that are not explained by Kaluza Klein: the yukawas, CKM, perhaps even part of the higgs structure. I would not try to use them as QFT gauge symmetries... but ok, you have not suggested any use at all, so perhaps you agree with me

20. Aug 8, 2012

### kneemo

To see the role of the non-compact exceptional groups in a geometric context, it's better to look at a simpler example.

Consider the complex projective plane, CP^2, for which SU(3) acts via isometries. If we would like to transform points in CP^2, and only care about preserving collinearity, we can use SL(3,C) transformations. In the Jordan algebraic context, SL(3,C) is the group of determinant preserving transformations of the Jordan algebra of 3x3 Hermitian matrices over C, J(3,C).

If we consider collineations (line preserving transformations) that fix a point in CP^2, these same transformations are those that preserve the determinant of 2x2 Hermitian matrices in the Jordan algebra J(2,C). These transformations lie in SL(2,C) and the homomorphism onto SO(3,1) follows pretty quickly once J(2,C) is identified with Minkowski spacetime such that the determinant is the squared length in (3,1) spacetime.

The relation between OP^2 and E6(-26) is basically just the octonionic version of that of CP^2 and SL(3,C), where SO(9,1) rather than SO(3,1) transformations fix a point in the plane. Using the split octonions, one has E6(6) with SO(5,5) transformation fixing a point in the "split" Cayley plane. The complex octonion (i.e. bioctonions) case yields E6(C) acting on the complexified Cayley plane (i.e. the bioctonionic projective plane) with SO(10,C) transformations fixing a point.

So, geometrically, E6 behaves more as a conformal group than an isometry group. Its use is more likely to be in the study of scattering amplitudes that are functions defined on the product of multiple copies of OP^2 (and its split and complex forms), with a copy for each particle.