# D=11 and almost predictive statements.

1. Jan 9, 2010

### arivero

It could be worthwhile to head a thread with some explanation about why D=11 is predictive, or perhaps to debate if it is.

First, we come to D=11 from first principles via two different ways: either by asking by an unification of D=10 string theories, or directly looking for the biggest supergravity theory. And here we can look dimensionwise or symmetrywise, because N=8 D=4 relates to D=1 sugra.

Now we look for symmetries of the 7 dimensional space.

The most symmetric thing we can find is

$$S^7$$

whose group of isometries is, of course $$SO(8)$$. It is a mathematical beauty, but it does not agree directly with Nature.

Now, part of the beauty of $$S^7$$ it that it has a Hopf fibering aspect,

$$\begin{matrix}S^3 \\ \downarrow \\ S^7 \\ \downarrow \\ S^4 \end{matrix}$$

And the same happens with $$S^3$$

$$\begin{matrix}S^1 \rightarrow S^3 \rightarrow S^2 \\ \downarrow \\ S^7 \\ \downarrow \\ S^4 \end{matrix}$$

This translates to a chain of subgroups
$$SO(8)\supset SO(4) \otimes SO(5) \supset U(1) \otimes SU(2) \otimes SO(5)$$

so that D=11 predicts almost the kind of symmetries we are going to find in the standard model. Except that it does not predict chirality, and it does not predict the (non chiral, by the way) SU(3) group. It is not even a subgroup of SO(5). But...

... the sphere $$S^4$$ has a peculiarity. It is the quotient under complex conjugation of the complex projective plane: $$S^4= CP^2 / O(1)$$. It could be said that really the seven-sphere is not the biggest seven dimensional manifold we can built, but that we need to add a discrete product (this complex conjugation, represented by the discrete O(1)) in the basis of the fiber bundle, so the whole construct is

$$\begin{matrix}S^1 \rightarrow L^3 \rightarrow S^2 \\ \downarrow \\ M^{pqr} \\ \downarrow \\ O(1) \rightarrow CP^2 \rightarrow S^4 \end{matrix}$$

... and now, note that the group of isometries of the complex projective plane is SU(3). So the group of isometries of any $$M^{pqr}$$ contains U(1)xSU(2)xSU(3). Actually,
it is exactly $$U(1) \otimes SU(2) \otimes SU(3)$$
except for two trivial constructs where it is enhanced to SO(4)xSU(3) or to SO(3)xSO(6).

So D=11 predicts the standard model.

Last edited: Jan 9, 2010
2. Jan 9, 2010

### arivero

Now, two questions I would like to find an answer in the literature, even if a footnote

1) Can we build a fiber bundle on S7 with fiber O(1), to produce an object with isometry group bigger than SO(8)?

2) Is there a classification of fiber bundles of Lens spaces (as fiber) over S4 (as basis)?

3. Jan 10, 2010

### arivero

I have been told that the relationship between S4 and CP2 is not well common, so I let me add a reference:

Last edited by a moderator: May 4, 2017
4. Jan 11, 2010

### arivero

This morning it dawned on me: the main families of 7 dimensional spaces which fiber in the way of the sphere are $$M^{pqr}$$ and $$N^{pqr}$$. The former have isometry group SU(3)xSU(2)xU(1), the later have isometry group SU(3)xU(1). A deformation or projection or whatever from M to N would be equivalent to the Higgs mechanism. So D=11 predicts the Higgs too?

5. Jan 13, 2010

### arivero

what the blip...

Let me review what do we know.

S4, S7, S13, according Atiyah and Berndt math/0206135 are quotients from projective planes CP2, HP2 and OP2 via O(1), U(1) and Sp(1). Of these, the first is discrete and it is the previously known Arnold-Massey-Kuiper theorem. To be noticed here:
• Arnold provides a different generalisation of the AMK theorem to spheres S1 (which is RP1) and S13 (whic is (HP4/AutH)/conj). He notes also that S/ is HP2/S1 etc. He also looks at S(3n+4)/O(4).
• Atiyah and Berndt use the embeddings of RP2 CP2 and HP2 inside S4 S7 and S13.
• Of course the groups O(1) U(1) and Sp(1) are the spheres S0 S1 and S3.

The following remark was communicated to Kreck and Stolz by M.F.Atiyah.
Kreck and Stolz use DNP notation instead of Wittens's. So $M_{k,l}$ is a S1 bundle over CP2xCP1 with first Chern class lx+ky, the xy being the generators of H2(Cp2) and H2(CP1). Or, the quotient of S5xS3 under a k,l action of S1. For coprime integers k,l one obtains all the 1-connected homogeneus manifolds with the Standad Model Symmetry. Iff k is even the manifold is spin.

6. Jan 13, 2010

### arivero

The other beast here are Aloff-Wallach spaces, the ones built as SU(3)/T1, usually labeled as $N_{k,l}$ According Kotschick and Terzic, remark 9, the usual definition excludes a case k=-l, and this case (where T1 is embedded inside the SU(2) embedded in SU(3)) is precisely the unique non trivial fibering by S2 over S5. In general, "they all have the real cohomology of S2 x S5". The article by K and T reviews the metrics on A-W spaces, and then it is interesting if we think this is the metric of the non chiral SU(3)xU(1) world.

The other new source of knowledge and folklore about these spaces is the 2002 article of http://arxiv.org/abs/hep-th/0108245" [Broken], which detached the topic from Kaluza Klein and reshaped it to modern "Maldacenian schools".
They first discuss "triaxially squashed three-spheres" to be used as fibers over S4, then producing more deformations of S7. Then Aloff-Wallach N(k,l) are reviewed and we are told a lot of scattered details.
• k=l is a SO(3) bundle over CP2
• N(0,1) and "S3-related" (1,0) and (1,-1) are special in that they only admit one Einstein metric, instead of two.
• N(k,l) is a S3/Zp lens space bundle over CP2, with p=|k+l|
• k=-l would produce "S3/Z0", the product S1xS2
• S3/Z2 is RP3 (=SO(3), as said above), so N(1,1) has such fiber over CP2, while N(1,-2) and N(-2,1) have fiber S3 (=SU(2)) . Still, they are related "corresponding to the action of the Weyl group S3 of SU(3)". The "cousins" of N(k,l) are N(k,-k-l) and N(-k-l,l)

CGLP notice the "Spin(7) manifold with Z2 orbifold singularity that one gets by replacing S4 by CP2 in the chiral spin bundle over S4" and they say that the principal orbits in this manifold are the N(1,1) example.

Thay also note that S5 (say, as needed by Kotschick and Terzic??) can be produced as the S1 bundle over CP2. The existence of only one non trivial S2 bundle over S5 is known to Steenrod, section 26.8.

By the way, the fact that S1 bundles over S2 are lens spaces appears also in Steenrod, 26.2 .

Last edited by a moderator: May 4, 2017
7. Jan 15, 2010

### Spinnor

Thank you for the interesting thread.

Would you please give a hand waving argument how this higher dimensional space gives rise to the chiral nature of the weak force or point me in the right direction.

Again, neat stuff, thanks!

8. Jan 15, 2010

### arivero

Let me try.

I think that first we solve a more easier question: how to give rise to the non chiral nature of the color+electromagnetic forces. We have two candidates from the point of view of kaluza klein:

-The 7 dimensional spaces $$N^{pqr}$$ (or N(p,q)), got from "SU(3)xU(1)/U(1)xU(1)" in the same way that Witten's spaces, or
-Their five dimensional descendants Npqr/S^2 (call them "SU(3)xU(1)/SU(2)xU(1)")

The first candidate has the advantage of being intensively studied in the eighties, and it could be connected to Witten Mpqr with a natural mechanism of symmetry breaking in the same way that the 7-sphere is broken to the squashed sphere.

The second candidate lives in D=9 and then it could support some of the string dualities. String duality is a topic relevant here because they connect chiral and not chiral theories either bouncing via D=9 or bouncing via D=11. This one is also interesting because 3 dimensional lens spaces L3 interpolate between S3 and S2xS1, and then their quotients L3/S2 interpolate between the interval and the circle (amusing).

In any case, if we understand SU(3)xU(1), we could be near of finding the mechanism to connect these spaces to the chiral version of Witten's $$M^{pqr}$$ and then to chiral SU(3)xSU(2)xU(1). Note that here the SU(3) is still non chiral. The SU(3) part comes from the symmetry of CP2 and then the argument about orbifolding CP2 down to the sphere could have a role. The SU(2)xU(1) come from the lens space fiber L3, and in the limit of S3 the symmetry automagically enhances to SU(2)xSU(2). So it seems that from the highest symmetry fiber bundle S3 --- > S7 ----> S4 Nature takes two counters:
- In the fiber S3, she prefers to squash to a generic L3.
- In the basis S4, she uplifts to its branched covering CP2.
Doing so, she reaches an object near to $$M^{pqr}$$ but where chiral fermions occur. But then they only appear to us broken again to non chiral SU(3)xU(1), ie to manifolds of the kind $$N^{pqr}$$ (or perhaps its descendants, as said above).

Amusingly, this way of thinking practically exhausts all the homogeneous manifolds of dimension seven.

9. Jan 17, 2010

### arivero

Re: what the blip...

A further remark: Hsiang Kleiner theorem tells that the only 4 dimensional 1-connected, strictly possitively curved closed Riemannian manifolds which admit an effective, isometric S1 action are S4 and CP2.

10. Jan 26, 2010