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Connes's noncommutative geometry program includes an approach to the Standard Model that employs a noncommutative extension of Riemannian metric. In recent years I've heard physicists say that this approach does not hold significant interest in the physics community.

Is this, in fact, the case? If so, why?

I do not mean for this question to be argumentative, but instead would like clarification.

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Related: – Qmechanic Nov 13 '12 at 15:13
Noncommutative geometry can have applications in M-theory, see here – Dilaton Nov 13 '12 at 15:53
@Qmechanic: The first question partially answers my question, I think... – Jon Bannon Nov 13 '12 at 18:46
In light of Qmechanic's link, and the fact that we now know the Higgs mass is a little less than 130, can the NCG picture be successfully modified to fit the actual Higgs mass? – Jon Bannon Nov 13 '12 at 20:07
If you are wondering about the use of noncommutative geometry in physics in general, I think that the most down-to-Earth use is probably Bellissard's use of NCG to explain the integer quantum hall effect. As far as I know, it is the only model of the integer quantum hall effect that takes into account all physical phenomena that are actually present. The Non-Commutative Geometry of the Quantum Hall Effect – Jon Paprocki Feb 8 '13 at 1:47

2 Answers 2

up vote 6 down vote accepted

The algebraic formulation of geometry as it appears in Connes's spectral formulation of geometry is in fact well-known elsewhere in physics, even if for some reason it is rarely highlighted as being the same (but see the references below): the way a spectral triple encodes a (non-commutative or classical) spacetime geometry is just the same mechanism by which a 2d superconformal field theory (a string worldsheet theory) encodes an effective target spacetime. This is not hard to see once one unwinds the definitions on both sides, but it is actually also a mathematically precise theorem (see again the references below).

This belated reply here is prompted by a talk that Alain Connes gave at our department yesterday, which reminded me of sitting down and writing a comment about this.

What Connes' NCG standard model construction really means.

For background on what the "Connes-Lott model" is about see the links here:

Connes likes to advertize (as he did again yesterday) his model as being a unification of gauge theory in geometry, since the gauge group becomes part of the diffeomorphism group of the ("slightly") non-commutative spacetime, and the gauge fields and Yukawa-couplings become part of the "internal" geometry.

But actually in itself that is not a new insights or achievement (I'll come in a moment to what IS a new insight here). Instead, that is precisely the old story of Kaluza-Klein compactification

But the following two points are interesting, albeit never mentioned in this context:

POINT 1. Traditional Kaluza-Klein compactification takes the compactification space to be a smooth manifold, and the big problem of the theory is that after compactification there are spurious effective fields -- the "moduli" --- which parameterize the small but finite Riemannian geometry of these smooth compactification spaces. The problem of "stabilizing" these moduli, hence to make the model be such that these moduli fields have mass outside the range of existing accelerator experiments, has been the huge topic in string theory in the last decade or two, culminating in that "landscape" business

What Connes' NCG model really does is observe that if one considers the degeneration limit where the 6-dimensional compactification space really collapses all the way to a point as a classical manifold, so that only a classically 0-dimensional space but non-classically (non-commutative) still with "KO-dimension" 6 remains, then also its spurious Riemannian moduli disappear.

That seems to be a neat little bit of insight, especially in view of...

POINT 2. Connes' spectral triples are naturally the point particle degeneration limit ("collapsing limit") of 2-dimensional superconformal field theories. That was maybe first understood in

Daniel Roggenkamp, Katrin Wendland, "Limits and Degenerations of Unitary Conformal Field Theories" (arXiv:hep-th/0308143) and "Decoding the geometry of conformal field theories" (arXiv:0803.0657)

and then put to use for the mathematical analysis of string theory in

Yan Soibelman, "Collapsing CFTs, spaces with non-negative Ricci curvature and nc-geometry", in Hisham Sati, Urs Schreiber (eds.), "Mathematical Foundations of Quantum Field and Perturbative String Theory", Proceedings of Symposia of Pure Mathematics volume 83 (2011)

There is also this more recent article of a very similar spirit, but using AQFT conformal nets alternatively to vertex operator algebras for describing 2d SCFT:

Sebastiano Carpi, Robin Hillier, Yasuyuki Kawahigashi, Roberto Longo, "Spectral triples and the super-Virasoro algebra", Commun.Math.Phys.295:71-97 (2010) (arXiv:0811.4128).

Namely the Hilbert space in the spectral triple is essentially the massless piece in the Hilbert space of the 2d SCFT, and the Dirac operator is the 0--mode of the 2d CFT's super-charge (the "Dirac-Ramond operator", that operator whose index is the Witten genus).

Now, 2d SCFTs of central charge 15 are precisely the "vacua" of string theory, and the way this works is precisely Connes's philosophy of spectral NC geometry (but enriched a good bit by higher modes): namely one thinks of the 2d SCFT as encoding the worldsheet quantum field theory of a superstring and thinks of it as encoding an effective spacetime geometry characterized "spectrally" as being whatever it is that this string "sees" as geometry, hence the geometry that may be recognized in its (energy) spectrum. In the point particle limit of the string this is exactly Connes's approach, both intuitively as well as mathematically.

So building spectral (potentially "non-commutative") models for particle physics is not actually Connes's new idea. That has been the idea of string theory all along, and what Connes considers is the point particle limit of that.

Now from that perspective his approach reads as follows: Let's build a systematic algebraic theory of point particle limits of string vacua, hence of 2d SCFTs. That's the theory of spectral triples. So then let's look there for those spectral triples which neatly encode an effective background spacetime that accomodates observed fundamental physics. One such is the Connes-Lott model.

The next natural question then would be: given that Connes algebraically found a neat point particle limit ("collapsing limit") of 2d SCFTs which fits well with observed physics, what can we say about lifting that back through the collapsing process to a genuine 2d SCFT ?

Phrased this way, it would seem to me that there should be some fruitful interaction between string phenomenology model builders and the group of people looking at Connes-Lott "non-commutative standard models". But there is not. My humble impression is that neither group has understood what the other group is really doing, that there is solid mathematics concretely relating the two.

And for one it is striking that Connes finds that the KO-dimension of his compact space has to be exactly 6, hence that he finds that the total KO dimension of his non-classical KK-spacetimes that yield the standard model has to be 10 = 4 + 6.

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This is a great answer! What you propose is certainly worth investigating. Hopefully there will be some work on your lifting problem. – Jon Bannon Mar 19 '14 at 20:24
clap clap clap! – user1504 Mar 19 '14 at 22:44
I just see that an earlier reference that highlights that the particle limit of the superstring's SCFT is a spectral triple are these lecture notes: Jürg Fröhlich, Krzysztof Gawędzki, "Conformal Field Theory and Geometry of Strings", extended lecture notes for lecture given at the Mathematical Quantum Theory Conference, Vancouver, Canada, August 4-8 . They explicitly spell out a bunch of examples. – Urs Schreiber Mar 20 '14 at 16:06

can the NCG picture be successfully modified to fit the actual Higgs mass?

arXiv:1208.1030 "Resilience of the Spectral Standard Model"

arXiv:1208.5023 "Asymptotic safety, hypergeometric functions, and the Higgs mass in spectral action models"

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Thanks, Dev. These papers certainly help. I'll accept the answer in light of the fact that the first paper seems to place NCG back in the game. – Jon Bannon Feb 10 '13 at 15:05

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