The 4th superstring revolution.

[arivero]

The spotting of the factor $${D-2 \over 24}$$ outside of the scope of string theory should be the trigger for a new revolution in this field. It could mean that a critical dimension is not exclusive of strings, or it could show how to avoid criticality after all.

The term itself is not rare, the denominator is a usual combinatorial factor, and D-2 can happen when we use the Riemann tensor in a general space.

In fact, it seems that the term can be made explicit by recasting Connes' fundamental theorem for commutative spectral triples, as it is done by Martinetti in th 2.11 of math-ph/0306046. By postulating this term to be equal to 1, we could get a critical dimension for spectral triples. Which is amazing, because spectral triples do not use strings inside its formulation.

 

[arivero]

Hmm Actually Martinetti already uses this explicit factor in his Ph D thesis, math-ph/0112038, theorem 1.21. Of course it depends on the normalization of the Wodzicki residue, which has been chosen differently of Connes 1996. I have not got here a copy of the handbook from Gracia-Bondia, Varilly and Figueroa. Could anyone to check how the theorem is formulated there?

 

[marcus]

The spotting of the factor $${D-2 \over 24}$$ outside of the scope of string theory should be the trigger for a new revolution in this field. It could mean that a critical dimension is not exclusive of strings, or it could show how to avoid criticality after all.
...

If I follow you, the D=26
in string arises by setting a certain factor equal to unity:

$$\frac{D-2}{24} = 1$$

And in what might hopefully turn out to be a useful introduction to the standard model from a noncommutative geometry standpoint, in which there is no string theory, the very same factor appears. slightly eerie.

http://arxiv.org/math-ph/0306046

A brief introduction to the noncommutative geometry description of particle physics standard model
Pierre Martinetti


"These notes present a brief introduction to Connes' non commutative geometry description of the standard model of particle physics. The notion of distance is emphasized, especially the possible interpretation of the Higgs field as the component of a discrete internal dimension. These notes are in french and are taken from the author's phD thesis."

You caution against getting too spooked by this since 24 is just
the factorial 4!
and there could be a simple explanation for the D-2.

Nevertheless it does whet the curiosity.

 

[arivero]

Well, the D-2 comes from the right place, namely a integration of the scalar curvature in a manifold by using the a Dirac structure on it.

The weak point is that Martinetti has chosen a "natural" normalisation for the Dixmier trace of the Dirac operator, such that in this normalization the "critical" factor is evident. The problem, as I see it, is to determine how natural the normalisation is. If it is, then the critical dimension of bosonic strings follows from there.

 

[arivero]

An example.

Suppose we want to expand perturbatively a gravity action using the Dixmier trace. Then each power will add a new factor (D-2)/24 which will make the Dixmier-based expansion more and more different from the naive expansion of the Einstein Hilbert integral. We would then suggest that only for D=26 the NCG and the naive relativistic expansion do coincide.

On the other side, we can suggest that the (D-2)/24 factor must be absorbed in the normalisation of the Dixmier trace (W. residue, to be right) and that NCG can avoid criticality. This should raise the question about if criticality in strings can equally be avoided.

 

[marcus]

This should raise the question about if criticality in strings can equally be avoided.

devil's advocate. they LIKE the criticality.
they always get mad when someone offers to cure them of it

 

[arivero]

Ah, that is the inconvenience of starting a revolution... one never knows what side will be shoot down.

 

[arivero]

Martinetti not guilty

I have spoken with Martinetti and he does not remember to have done any notational change. So the main suspect now is the book of Varilly, Gracia-Bondia and Figueroa, which regretly is the one I have not access to, just now... and I do not forsee to be able to get it for a couple months.

 

[arivero]

Tha blame is finally upon GraciaBondia-Varilly-Figueroa, a book that is to NCG as Polchinski is to strings. There, in theorem 11.2, page 492, they acknowledge the change of normalisation, on grounds of consistency with the general non-commutative integral.

 

[arivero]

I historical account of this factor has been uploaded to the ArXiV, of course in its more speculative section, http://arxiv.org/abs/physics/0409022

 

[marcus]

Congratulations are called for, AR,
it's a nice paper! Thanks for posting the link.

 

[arivero]

Thanks marcus. Let me to expand in the last part of the upload. The old aspiration of string theory was that each mode of the string should give rise to a different particle. The bosonic string has 24 transversal modes available in the m=0 excitation (also it has one m<0 excitation, the tachion, but this was cured in the heterotic string).

If we look at the SM experimental input looking only for mass eigenstates, no for charge eigenstates, we find 12 mass subsectors in the fermion side and 12 bosons in the gauge side. To me, this only point of strings is more important that the buzz about if they have or they have not got gravity.

The heterotic strings lives with a leg in the 10 dimensional world and another in the 26 dimensional one, so a decent arrangement of the particle content should still have 24 elementary transversal directions. This arrangement is complicated because they have a lot of freedom. They can convert between bosons and fermions via bosonization (a phenomena of the 2D worldsheet) and the can invoke supersymmetry to look for partners of a given boson or fermion.

M-theory (and F-theory) have a possibility to score because of the Higgs. We know that Higgses can be related to extra dimensions adjoined to ST. Now, ¿how many? In toy models, only one. But a Higgs doublet has four real fields, so it could go up to 4. Two Higgs doublets, as in most SUSY models -and also my inquiry into nuclear physics- could add up to 8 extra dimensions, doing a 12 dimensional ST the, er, natural scenario.

I do not believe that these dimensions are to be compactified; they should be quotiented out, perhaps using the concept of groupoid from Connes and Morita equivalence. This process could let us with the discrete part of Connes-Lott models.