## September 6, 2006

### Connes on Spectral Geometry of the Standard Model, II

#### Posted by Urs Schreiber

Last time I made some general remarks on the idea of spectral triples and of action functionals obtained from them.

Here I make some general remarks on the nature and the implication of the results found by Connes, in his search for the spectral triple describing our world.

I tried to point how a spectral triple is just another way to look at smooth functors from super-worldlines to graded Hilbert spaces, hence something very natural with respect to the doctrine of quantum mechanics.

By forming some obvious generalization of a heat kernel expansion of a given spectral triple, each such triple determines a functional which looks like the sort of action functionals that physicists usually write down in order to describe theories involving, gravity, gauge forces, Higgs bosons and fermionic matter.

Turning this around, and postulating that the action functionals of our physical theories should be the heat kernels of given spectral triples, is known as the spectral action principle.

The task is hence to find precisely that spectral triple whose generalized heat kernel yields the action functional describing our best knowledge about the fundamental constituents of the experimentally tested world, namely the action of the standard model of particle physics coupled to gravity.

That it is rather easily possible to find spectral triples that get very close indeed to this goal has been demonstrated early on. The canonical review is

Ali H. Chamseddine, Alain Connes
The Spectral Action Principle
hep-th/9606001 .

While already impressive, there were two small details which did not quite work out.

• It was pointed out in hep-th/9610035 that the model suffered from something called fermion doubling, usually an artefact of a coarse-grained description that is encountered in lattice approximations in gauge theory.
• The original spectral triple used did not reproduce certain information about neutrino mixing and neutrino masses. (Probably because these were found experimentally only after that model was developed, I assume.)

As has been reported elsewhere already, Alain Connes is currently giving talks about recent progress that has been made on these issues,

Alain Connes
Noncommutative Geometry and the standard model with neutrino mixing
talk at Isaac Newton Institute Workshop Noncommutative Geometry and Physics: Fundamental Structure of Space and Time
(abstract),

a summary of which has appeared on the arXiv:

Alain Connes
Noncommutative Geometry and the standard model with neutrino mixing
hep-th/0608226.

Not only have the above two problems been resolved, but it also turns out that some of the choices that have to be made to specify the spectral triple could be “unified” in a more elegant description.

What is hence the result that is obtained here?

It’s not that curious quantitative details of the standard model, like the precise values of coupling constants or the number of generations of fermions can be predicted.

In order to reproduce these, one has to choose “by hand” the appropriate $C^*$-algebra of the spectral triple and its representation, and one has to choose an appropriate “metric” on the “internal space”, namely an appropriate generalized Dirac operator, essentially given by the Yukawa coupling matrix.

One could argue, though, that the spectral action principle successfully predicts the qualitative features of the physical world we observe, namely the presence of gravity coupled to Yang-Mills-like gauge forces. As Connes remarks in the introduction of his latest paper, there is no choice involved in the nature of the “generalized heat kernel”, and all those heat kernels have an expansion that happens to start with the Einstein-Hilbert action coupled to Yang-Mills-like interactions.

Of course you may complain that this is simply because the curvature tensors that these theories are built from happen to be the only possible terms that we have a right to expect to appear in a heat kernel expansion. But still, I think, it is a remarkable fact that all this works out as it does. In the same vein you could say that what Einstein/Hilbert and Yang/Mills achieved is just noticing the obvious, namely the first order scalar invariants that can be written down for a connection. Still, it is an achievement to see that this is the principle we need for physics.

But there are two main implications which I think are important about Connes’ result

• A spectral Riemannian geometry completely and uniquely specifying the “external and internal” geometry of the observable world does exist. On top of that, it’s algebraic formulation in terms of spectral triples is tremendously more elegant and compact than the standard way to code these things into symbols. If you believe in a connection between elegance and truth, this is something to take notice of.
• Even better, since it does exist and has been identified, we may investigate special geometric properties this geometric description of our world has. For instance we may compute its spectral dimension. In spectral geometry there are in fact various notions of dimension, reflecting the fact that algebraic geometry is more general than ordinary geometry. There is something called the “metric dimension” and something called the “KO-dimension”. Connes shows that

If you regard gravity as the curvature of spacetime (which we long ago learned is right) and if you furthermore regard the coupling constants of particles in our world as the metric of an “internal”, “compactified” part of spacetime (which now Connes shows is a viable point of view in the context of spectral geometry) then it is a fact that the experimentally observed metric dimension of our experimentally accessible world is $d_\text{metric} = 4 \in \mathbb{Z}$ while something called the “KO-dimension” of the accesible part of the world is $d_\text{KO} = 4+6 = 10 \in \mathbb{Z}_8 \,.$

The KO-dimension of an algebra should be something like the dimension as seen by the real algebraic K-theory (I, II) of that algebra, I guess. The definition is apparently in

Alain Connes
Noncommutative geometry and reality
J, Math. Phys. 36, 11 (1995) pp. 6194-6231
(pdf),

but currently, for some reason, I am having problems displaying the pdf behind this link.

If you accept the generalization from ordinary to spectral geometry, then Connes has produced the first realistic Kaluza-Klein reduction.

Since this may sound drastic to some people, let me reformulate this slightly.

In physics, it has become customary to call models of spacetime which involve generalization of ordinary geometry to non-commutative or otherwise algebraically formulated concepts non-geometric phases of spacetime.

Then:

Connes has found a realistic Kaluza-Klein compactification with the compact space being in a non-geometric phase.

It is hard not to notice the fact that perturbative superstring theory asserts that the world is a Kaluza-Klein reduction of “cc-dimension”

(1)$d_\mathrm{cc} = 4 + 6 = 10$

on a possibly non-geometric $(d_\mathrm{cc} = 6)$-dimensional internal space.

In perturbative string theory, geometry is not determined by a spectral triple, but by something like a 2-spectral triple, if you allow me to use this terminology.

Where a spectral triple is more or less the same as a smooth functor from super-worldlines to graded Hilbert spaces

(2)$1\mathrm{SCob} \to \mathrm{SHilb}$

a 2-spectral triple in my sense here is a functor on super-worldsheets to super-Hilbert spaces

(3)$2\mathrm{SCob} \to \mathrm{SHilb} \,.$

(I am not indicating it in my notation here for simplicity, but in the first case we require a super-Riemannian structure on our cobordisms, while in the second we just require a super-conformal structure.)

Such functors have a characteristic number associated to them, called their conformal central charge. The number $d_\mathrm{cc}$ I mention above is $2/3$ times this central charge.

As one can see on P.P. Cook’s blog entry, this obvious coincidence of the numbers $4+6$ has not escaped notice. Apparently some people, including Connes, talked about whether or not Connes’ internal $(d_\mathrm{KO}= 6)$-dimensional space could describe a “nongeometric phase of a Calabi-Yau space”.

I think clearly it can not, since if it were, then the corresponding spectral action should be supersymmetric, which it is not. But I see no reason why it could not be a non-geometric phase of a more general $d_{\mathrm{cc}} = 6$ compactification.

Certainly, speaking about 2-spectral triples is very suggestive. But it has not been really worked out yet at all. The closest (except for my own dreams about it) that I have seen, is the work that Soibelman reported on at a conference in Vienna this summer ($\to$) as well as parts of what Stolz&Teichner discuss in their

S. Stolz & P. Teichner
What is an elliptic object?
(pdf)

aspects of which, relevant to the present discussion, I tried to review here.

But it might be noteworthy to recall that Ali Chamseddine, Connes’ collaborator on the spectral action principle research, once began trying to compute the 2-spectral action of the 2-Dirac operator found in string theory

A. H. Chamseddine
The Spectral Action Principle in Noncommutative Geometry and the Superstring
hep-th/9701096

Ali H. Chamseddine
An Effective Superstring Spectral Action
hep-th/9705153 .

He claims to reproduce parts of the well-known effective target space action of string theory this way.

Related work along these lines is for instance

David D. Song, Richard J. Szabo
Spectral Geometry of Heterotic Compactifications
hep-th/9812235 .

So I think this should make us want to try to find a $c=9$ SCFT whose point particle limit combined with the standard $c = 6$ SCFT describing a 4-dimensional geometric target space reproduces the spectral triple found by Connes. Of course people have been doing precisely that for quite a while.

Volker Braun, Yang-Hui He, Burt A. Ovrut, Tony Pantev
The Exact MSSM Spectrum from String Theory
hep-th/0512177 .

I am not into this model building, so I cannot quite tell. But maybe Connes’ result might help to approach this model building from the other side. Instead of writing down SCFTs (2-spectral triples of sorts) and checking if they reproduce the standard model in a suitable 1-spectral triple limit, Connes writes out the standard model itself in the required 1-spectral form. Maybe that helps. Maybe not.

Posted at September 6, 2006 3:53 PM UTC

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### the parameters of the standard model

I thought I’d post a third entry with technical details, but certainly not today.

But since several people that I have talked to wondered about it, I want at least quickly point out how the famous dimensionless parameters of the standard model are encoded in the spectral triple.

- the Yukawa couplings are encoded in the geometry of the internal space, namely in the Dirac operator corresponding to the internal space. In fact, that Dirac operator is more or less nothing but the matrix of Yukawa couplings. See equation (1.21) in the Chamseddine-Connes review # (and again later between equations (3.4) and (3.5).)

So: it’s the metric of the “generalized compact internal space that we have KK-reduced on”, which determines the Yukawa couplings.

[Update (7. Sept. 06): My next statement here is nonsense. See below.]

- the (running) gauge couplings are determined by the volume of spacetime (!) apparently, together with the details of the “cutoff function” $f$ that enters the definition of the bosonic part of the spectral action

(1)$\mathrm{Tr}(f(D/\Lambda))$

(beware of slight difference in the choice of symbols in the old review and the new paper).

Equation (2.20) defines the quantity

(2)$a_0 = \frac{N}{2\pi^2} \int_M \sqrt{g} d^4 x$

which is related to the bare gauge coupling $g_0$ in equation (2.29)

(3)$a_0 = \frac{-3N}{80}\frac{1}{g_0^2} \,.$
Posted by: urs on September 6, 2006 8:30 PM | Permalink | Reply to this

### Re: the parameters of the standard model

I think you might be misinterpreting; his a0(P) seems to contribute the cosmological constant term in 2.26, while the C2 term is from a4(P). (See equation 2.24; the coefficient of C2? there doesn’t seem to depend on the volume of space.) The “a0” in equation 2.28 seems to not mean the earlier a0(P), but to just be a constant.

(On the other hand, maybe I’m misinterpreting, but it would be very weird if the gauge coupling scaled in this way. For one thing, dimensional analysis gets all screwy. And the equations in the paper don’t seem to me to suggest your interpretation.)

Posted by: anon. on September 7, 2006 2:10 AM | Permalink | Reply to this

### Re: the parameters of the standard model

The “$a_0$” in equation 2.28 seems to not mean the earlier $a_0(P)$, but to just be a constant.

Yes, right, thanks for catching that! That was nonsense what I wrote.

Instead, the bare gauge coupling is part of the data determining the Dirac operator, of course, as it should be. It appears in equation (2.4).

Ok, good. So indeed all the dimensionless parameters are encoded in the metric of the spectral compactified geometry.

Posted by: urs on September 7, 2006 11:58 AM | Permalink | Reply to this

### Re: the parameters of the standard model

all the dimensionless parameters are encoded in the metric of the spectral compactified geometry.

Remaining question: how does the Higgs come in?

It’s the part of the connection “with spacetime index along the compactified space”, in a sense. That’s described on p. 26 of hep-th/9603053.

Posted by: urs on September 7, 2006 8:10 PM | Permalink | Reply to this

### Re: Connes on Spectral Geometry of the Standard Model, II

Could you explain more about whether/how the Connes geometry can be thought of as the target space of a CFT?

thanks!

Posted by: boreds on September 6, 2006 9:41 PM | Permalink | Reply to this

### Re: Connes on Spectral Geometry of the Standard Model, II

Could you explain more about whether/how the Connes geometry can be thought of as the target space of a CFT?

I only remarked that we have two approaches that share some remarkable structural similarities.

1) In one approach, you sort of look at a superparticle, whose dynamics is given by some Dirac operator $D$. You associate to that operator an algebraic notion of dimension, which happens to be $d_\mathrm{KO} = 4 + 6 \in \mathbb{Z}_8$ for phenomenologically realistic choices of $D$.

2) On the other hand, you start with a superstring, governed by something like a “2-Dirac operator” $G$ (the Dirac-Ramond operator). Again, you associate an algebraic measure $d_{cc}$ of dimension to this setup, this times defined as two thirds of the conformal central charge. And it turns out that choices of $G$ that are to have a chance to yield anything phenomenologically realistic are obtained for $d_{cc} = 4 + 6$.

There is a clear analogy. I think it is natural to guess that – or at least to ask if – 1) can be realized as some limit of 2).

That’s all I know at the moment.

Posted by: urs on September 7, 2006 12:11 PM | Permalink | Reply to this

### Re: Connes on Spectral Geometry of the Standard Model, II

A long while ago, somebody called boreds wrote

Could you explain more about whether/how the Connes geometry can be thought of as the target space of a CFT?

There is a better answer to this than I gave at that time – and I am not sure why I didn’t mention it back then.

Anyway, the important point is that:

there is some sort of point particle limit for 2d CFTs in which they reduce to Connes’ spectral triples.

It’s not supposed to be surprising, because a 2-d CFT can be regarded as quantum mechanics on loop space over target space, in a way, and in the limit that the loops tend to constant loops, this plausibly tends to quantum mechanics on target space. Finally, a spectral triple is nothing but the data of a (supsersymmetric) quantum mechanical system.

Luckily, on top of these general vague remarks, there is detailed work on this 2dCFT$\to$ spectral triple limit:

Daniel Roggenkamp, Katrin Wendland, Limits and Degenerations of Unitary Conformal Field Theories

See in particular paragraph 1.2 “Spectral Triples from CFTs”, starting on p.6 .

A few remarks on that, once taken in a talk by Yan Soibelman, can be found here.

Posted by: urs on February 14, 2007 9:34 PM | Permalink | Reply to this

### Re: Connes on Spectral Geometry of the Standard Model, II

Any comments on the John Barrett paper?

A Lorentzian version of the non-commutative geometry of the standard model of particle physics

Best wishes,

Posted by: PPCook on September 7, 2006 4:46 PM | Permalink | Reply to this

### Re: Connes on Spectral Geometry of the Standard Model, II

That’s interesting, judging from the comment at the very end Barrett and Connes et al. arrived at this independently and put the report on the arXiv almost simultaneously.

I have just had a first look:

John Barrett gives a very clear discussion of the issue, explaining the former doubling problem (really a quadrupling problem), its obvious solution and how that leads one from $d_\mathrm{KO} = 0$ to $d_\mathrm{KO} = 6$.

Posted by: urs on September 7, 2006 5:49 PM | Permalink | Reply to this

### Re: Connes on Spectral Geometry of the Standard Model, II

This Monday Barrett has done a very detailed presentation of his work. When you build carefully J and D in the Minkowski space, the solution seems obvious :-) Connes stressed the fact that this aproach actually gives a role for J and gamma: they impose the majorana condition and control the doubling (quadrupling) of fermions.

The question of the nomenclature (dimension or signature) was not a hard one nor a fundamental thing. After the talk, I asked about it and Barret pointed me towards “The Spinorial Chessboard” but I have noticed that also Lawson-Michelsohn covers it: just see the (1,1) periodicity theorem for KR-theory, around page 72.

Posted by: Alejandro Rivero on September 11, 2006 6:47 PM | Permalink | Reply to this

### Re: Connes on Spectral Geometry of the Standard Model, II

This Monday Barrett has done a very detailed presentation of his work. When you build carefully $J$ and $D$ in the Minkowski space, the solution seems obvious :-) Connes stressed the fact that this aproach actually gives a role for $J$ and gamma: they impose the majorana condition and control the doubling (quadrupling) of fermions.

This sounds like precisely the argument that Barrett gives in his paper.

So I conclude from your description that me may trust that Barrett’s paper is the official state-of-the-art then?

The question of the nomenclature (dimension or signature) was not a hard one nor a fundamental thing.

Could you please remind me which question precisely you are referring to? Thanks!

Posted by: urs on September 11, 2006 7:07 PM | Permalink | Reply to this

### Re: Connes on Spectral Geometry of the Standard Model, II

bq. The question of the nomenclature (dimension or signature) was not a hard one nor a fundamental thing.

Could you please remind me which question precisely you are referring to?

It was amusing to read Barrett speaking about signature while Connes speaks of KO dimension. The signature point of view is also a bit more interesting; for instance some (Majid?) pointed out that usual 1+3 space time can be said to have also signature 6 mod 8 (it is -2, and mod 8…)

And yep, it seems to me that Barrett is co-oficial state of art now; still lacking to check how the Minkowskian and Euclidean approaches are related. Barrett gets its commutative part of the spectral triple straightly from physics, and this gives some confidence that if some axiom is violated, could be time to blame the axiom.

Posted by: Alejandro Rivero on September 12, 2006 9:37 AM | Permalink | Reply to this
Read the post Connes on Spectral Geometry of the Standard Model, III
Weblog: The n-Category Café
Excerpt: First technical details for spectral action theory.
Tracked: September 7, 2006 4:58 PM
Read the post Connes on Spectral Geometry of the Standard Model, IV
Weblog: The n-Category Café
Excerpt: The details of the spectral triple which describes the standard model.
Tracked: September 8, 2006 2:33 PM

### coleman mandula.

What is strange about Connes product of spectral triples, and perhaps of a good bunch of Kaluza Klein reductions, is that the masses live in the internal geometry. We are very well used to the product of Poincare times Internal symmetries, and we know that mass and spin label the Poincare part, charges label the internal part. But here the internal part has a Dirac operator on its own, carrying the data of mass and spin. It is amazing.

For instance, it means that the tensor product of the finite spectral triple times itself builds an object carrying information of the constituient masses, then more alike (compared to the usual product of representations) to mesons, positronia, or diquarks but still without spatial information.

Posted by: Alejandro Rivero on October 26, 2006 7:55 PM | Permalink | Reply to this
Read the post QFT of Charged n-Particle: T-Duality
Weblog: The n-Category Café
Excerpt: Topological T-duality as a pull-push transformation of sections of the 2-particle.
Tracked: February 16, 2007 2:17 PM
Read the post Report-Back on BMC
Weblog: The n-Category Café
Excerpt: Bruce Bartlett reports from the British Mathematics Colloquium 2007
Tracked: April 22, 2007 8:18 PM
Read the post Spectral Triples and Graph Field Theory
Weblog: The n-Category Café
Excerpt: Yan Soibelman is thinking about spectral stringy geometry.
Tracked: June 12, 2007 9:01 PM

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