New papers on NCG and the standard model

[arivero]

Well, not new insights, but at least new paper and a new face:

http://arxiv.org/abs/1204.0328 Particle Physics from Almost Commutative Spacetimes by Koen van den Dungen, Walter D. van Suijlekom

the veteran in the team is van Suijlekom and it seems that he is pushing forward with some strength. Recent papers show interest for the renormalization process in NCG and http://arxiv.org/abs/1003.3788 was the first new try towards SUSY.

I notice also that the long struggle of Connes and Chamseddine to understand the relationship between the SM algebra and the Pati Salam algebra has at least got some other groups involved. For instance http://arxiv.org/abs/1112.3622 do mention both algebras in the abstract.

Connes and Chamseddine pursue the uniqueness of Pati Salam in a series of papers. It appears as theorem III.1 of http://arxiv.org/abs/0706.3690 and then it is mentioned again in section 5 of http://arxiv.org/pdf/1008.3980v1.pdf ( where they make an attempt to understand how to trade orientability by chirality) and in graphical form in figure 2 of http://arxiv.org/pdf/1008.0985v1.pdf

 

[arivero]

Let me review what is going on. During the work of Connes Chamseddine and Marcolli, in 2006, it emerged that the standard model algebra was really extracted from the Pati Salam algebra.

In a more concrete way, http://arxiv.org/abs/0706.3690 ask to search the algebras which are compatible with some elementary requisites, plus a first order condition and a non degeneracy condition that amounts to impose to the spectral triple a KO-dimension of 6 mod 10. These conditiones uniquely select the algebra H+H+M_4(C), generatiog thePati-Salam group. From it, the Standard Model algebra C+H+M_3(C) appears as its even part (IIRC; in any case it is a unique way).

Now, Pati-Salam and its production of the SM group are well known in the theory of extra dimensions. The later is the invariance group of the 8-dimensional manifold S3xS5, and Witten found that any quotient of this manifold by a U(1) action (or a S1 map if you prefer) will produce a 7-dimensional manifold whose invariance group is the SM group U(1)xSU(2)xSU(3).

So what is amusing is that Connes seems to be doing all the game in six extra dimensions, while the classical geometry interplays between 7 and 8. All all of you are well aware that six extra dimensions is the requisite of string theory, while seven is M-theory and its cousin, maximal supergravity.

Of course, it could be a red-herring; back in 2006 it was also spoken that 10 mod 8 could be simply the signature of space time (3-1=2, which is also equal mod 8 to ten), but any game with signatures asks to understand the process of going from Euclidean to Minkowski in Connes models.

 

[mitchell porter]

I have not been a fan of Pati-Salam before, and for actual phenomenology I'm currently looking at SU(5); but the most appealing string model that I have recently examined obtains Pati-Salam from branes at a "del Pezzo" singularity. The paper is http://arxiv.org/abs/1106.6039; Lubos blogged about it. The heart of the model, as illustrated on page 7 of the paper, is really elegant; it becomes much less so (page 9, page 14) when they start hacking with it in order to match all the standard model details, though it's still impressive that they can do so at all; and in an appendix they mention an amended starting point which might not require so many attachments to become realistic.

The way Pati-Salam is obtained here: You have three large dimensions times a six-dimensional "cone" with a specific five-dimensional cross-section (see page 6 here). You then have D3-branes resting in the three-dimensional submanifold consisting of the three large dimensions times the zero-dimensional point at the tip of the cone, and the strings come from the local fluctuations of the branes "up" the local copy of the cone. So this has nothing to do with pre-string Kaluza-Klein; I can't say for sure whether it has anything to do with this NCG Pati-Salam, which might conceivably be a truncation, but probably it's unrelated.

 

[friend]

I notice also that the long struggle of Connes and Chamseddine to understand the relationship between the SM algebra and the Pati Salam algebra has at least got some other groups involved. For instance http://arxiv.org/abs/1112.3622 do mention both algebras in the abstract.

Is this in any way related to the effort to derive the SM from quaternions and octonions, which are hypercomplex numbers? These hypercomplex numbers are a sub-algebra of the Clifford algebra that is also use as an alternative to the differential geometry used in GR. If I understand it correctly, the Clifford algebra specifies a non-commutative relationship between the basis vectors of the curved, background spacetime. So all this sound like the same kind of effort as above.

 

[arivero]

Is this in any way related to the effort to derive the SM from quaternions and octonions, which are hypercomplex numbers?

Yes, it is related to the quaternions and octonions, but not to "the effort" because there is not such effort, only scattered attempts. Everybody in the mainstream seems to know that quaternions and octonions are deeply in the mathematical structure, but it is commonly regarded as an unproductive knowledge, not driving to further advance :-(