It could be worthwhile to head a thread with some explanation about why D=11 is predictive, or perhaps to debate if it is.(adsbygoogle = window.adsbygoogle || []).push({});

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

[tex]S^7[/tex]

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

Now, part of the beauty of [tex]S^7[/tex] it that it has a Hopf fibering aspect,

[tex]\begin{matrix}S^3 \\ \downarrow \\ S^7 \\ \downarrow \\ S^4 \end{matrix}[/tex]

And the same happens with [tex]S^3[/tex]

[tex]\begin{matrix}S^1 \rightarrow S^3 \rightarrow S^2 \\ \downarrow \\ S^7 \\ \downarrow \\ S^4 \end{matrix}[/tex]

This translates to a chain of subgroups

[tex]SO(8)\supset SO(4) \otimes SO(5) \supset U(1) \otimes SU(2) \otimes SO(5) [/tex]

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 [tex]S^4[/tex] has a peculiarity. It is the quotient under complex conjugation of the complex projective plane: [tex]S^4= CP^2 / O(1)[/tex]. 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

[tex]\begin{matrix}S^1 \rightarrow L^3 \rightarrow S^2 \\ \downarrow \\ M^{pqr} \\ \downarrow \\

O(1) \rightarrow CP^2 \rightarrow S^4 \end{matrix}[/tex]

... and now, note that the group of isometries of the complex projective plane is SU(3). So the group of isometries of any [tex] M^{pqr}[/tex] contains U(1)xSU(2)xSU(3). Actually,

it is exactly [tex]U(1) \otimes SU(2) \otimes SU(3)[/tex]

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.

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# D=11 and almost predictive statements.

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