Connes-Chamseddine: Math Evidence for Kaluza-Klein Theory

[arivero]

I got some spare time to glance at "[URL
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Section 9.2 is awesome. It tells us that the discrete algebras of NCG should be interpreted as a low energy residual of a NC deformation of an extra dimensional manifold. For instance, a single M_4(C) is a hint of a deformation of the sphere S4.

Basically, Connes and Chamseddine are reporting now a mathematical three-sigma experimental evidence for kaluza klein theory.

1-sigma: red book NCG, Connes-Lott see http://dftuz.unizar.es/~rivero/research/ncactors.html
2-sigma: 6 dimensional spectral triple, Connes-Chamseddine-Marcolli 2006
3-sigma: Connes-Chamseddine 2010
4-sigma: production of the manifold related to M_2(H)+M_4(C).
5-sigma: ?

Let me note that it could be a six dimensional manifold related to a 8 dimensional one, or reciprocally, because we know that:
- Pati Salam needs 8 extra dimensions (same than other L-R models, btw).
- Standard Model unbroken needs 7 extra dimensions.
- a lot of people needs 6 extra dimensions.
- SU(3)xU(1) EM (the unbroken group of the SM) can live in 5 extra dimensions (same than pure SU(4), btw, because su(4)=so(6)).

 

[marcus]

I got some spare time to glance at "[URL
[/URL]
Section 9.2 is awesome.

I didn't post because I was hoping someone like Hans or Larsson would respond. I think the insight here is yours and you should write it up for arxiv. The idea of KK-residue, the low-energy diminished remnant of KK symmetry. I do not see it clearly presented that way the authors in their section 9.

Thanks for calling attention to C.C.'s paper. I was impressed by the predictions listed at the beginning of section 9.

 

[arivero]

. The idea of KK-residue, the low-energy diminished remnant of KK symmetry. I do not see it clearly presented that way the authors in their section 9.

Hmm, ok, they say that "This leads us to investigate the postulate that at very high energies, the structure of space time becomes noncommutative in a nontrivial way" and that "it is natural to modify the basic assumption we made that space-time is a product of a continuous four dimensional manifold times a finite space." But then they do not postulate straighfowardly a KK structure, they only suggest that "A good starting point would be to mesh in a smooth manner the four-dimensional manifold with the finite space $$M_2 (H) \oplus M_4 (C)$$".

But note that the suggestion is done after showing two examples where:
- the mesh of a single symbol with a finite space $$M_2(C)$$ happens to produce a 2 dimensional space, and
- the mesh of a single object with a finite space $$M_4(C)$$ happens to produce a 4 dimensional space.

Now, is the dimensionality of the final space determined by the symbol, by the matrix, or by a consistency condition between them? What is the dimension we expect if instead of using $$M_4(C)$$ we use $$M_4 (C) \oplus M_4 (C)$$? or $$M_2 (H) \oplus M_4 (C)$$? This is the kind of questions that seem obvious after reading 9.2, and they point to spaces whose dimensionality is greater that the ones in the examples.

(this parragraph is certainly composed by conjectures of mine:) I would expect dimension 8 because $S^5 \times S^3$ has the isometry group $SO(6) \times SO(4)$, which happens to be locally as $SU(4)\times SU(2)\times SU(2)$. But the KO space of Connes is dimension 6, and that should induce a breaking: First down to dimension 7, where $(S^5\times S^3) / U(1)$ actually has the isometry of the unbroken standard model, and then down to dimension 6 and massive W, Z bosons. Note that in KK there is also another nice 8 dim candidate, $CP^2\times S^1\times S^3$. And note another conjecture of mine, that taking W and Z mass to infinite we go down to dimension 5, with space $CP^2\times S^1$.