## September 8, 2006

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

#### Posted by Urs Schreiber

With some background material in place # # # I’ll now try to indicate how we can get something like the standard model as the effective target space theory of our “superparticle”.

(I’ll stick to this superparticle imagery #, but you should remember that this is my prose around Connes et al.’s formulas.).

We need to find the right compactification geometry that will produce an effective standard model on the compactified space.

(I’ll no longer distinguish between “ordinary spaces and geometry” and “generalized spectral spaces and geometry”. Similarly, I’ll no longer distinguish between “metric” and “Dirac operator”, nor between “topological space” and “algebra”, and so on. We are now firmly in the quantum world, where spacetime is nothing but the world as seen by our quantum superparticle.)

The first proposal for such compactifications, which already came very close to the standard model, is decades old by now.

Alain Connes & John. Lott
Particle Models And Noncommutative Geometry
Nucl.Phys.Proc.Suppl.18B:29-47,1991
(spires).

See for instance the review

Daniel Kastler, Thomas Schucker
The Standard Model a la Connes-Lott
hep-th/9412185 .

As far as I can tell, at that time the Yang-Mills terms were stilll included by hand. The main point of the spectral approach was to realize that it could nicely explain the Higgs boson and its Yukawa coupling terms to the fermions. It did (and does) so by realizing the Higgs boson as an internal part of an ordinary minimally coupled connection 1-form - an internal gauge boson.

A few years later Connes apparently realized that also the gravitational and gauge kinetic Yang-Mills terms had an inherent operator-theoretic formulation, namely the spectral action principle.

This is indicated in

A. Connes
Gravity coupled with matter and foundation of non-commutative geometry
hep-th/9603053

and fully formulated in

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

both of which start by presenting the general spectral idea and then working out how to realize the standard model in detail.

I find that a very useful discussion of the details of the realization of the standard model in this approach is

C. P. Martin, Jose M. Gracia-Bondia, Joseph C. Varilly
The Standard Model as a noncommutative geometry: the low energy regime
hep-th/9605001 .

This is in particular designed to take the ordinary physicst by the hand and introduce him or her gently to the operator-theoretic spectral description by motivating these by the structure of the standard model.

So already over ten years ago people had a pretty good idea that and how the standard model action has an elegant descritption as a spectral action.

The only problem was: this description was wrong.

But only in the sense in which ideas in physics tend to be wrong - not entirely wrong but not quite right.

Namely, instead of containing the fermionic particle content of the standard model, these spectral models produced four copies of every fermion in the standard model. A slight overkill.

By a remarkable synchronicity, it seems that there was no progress on this aspect for about ten years, and now two preprints appear almost simultaneously, presenting the solution:

John W. Barrett
A Lorentzian version of the non-commutative geometry of the standard model of particle physics
hep-th/0608221.

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

And the modification needed to get this solution is rather tiny, just a small change in the real structure $J$ of the spectral triple from ten years ago.

All right, I should finally say now what this compactification geometry is. Up to slight technical details that I will get wrong, it’s this:

First of all, the topology of our 4+6-dimensional compactified space is given by the algebra of functions

(1)$A = A_\mathrm{ext} \otimes A_\mathrm{int}$

on it, where

(2)$A_\mathrm{ext} = C^\infty(X)$

is simply the algebra of smooth functions on a 4-dimensional spin manifold $X$ and

(3)$A_\mathrm{int} = \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$

is the algebra “of functions on the internal space” $F$, where

(4)$\mathbb{C}$

is the algebra of complex numbers

(5)$\mathbb{H}$

is the algebra of quaternions and

(6)$M_3(\mathbb{C})$

that of $3\times 3$ matrices with entries complex numbers.

Sometimes people call this an almost commutative algebra. This is in order to distinguish it from the case where also the algebra $A_\mathrm{ext}$ of functions on the observable macroscopic space is non-commutative, which might be for instance true if you look at 4-dimensional space very closely, at extremely small scales.

Of course this choice of algebra is related to the gauge group of the standard model. In the algebraic approach, the natural group to study is the group

(7)$\mathrm{Aut}(A)$

of automorphisms of the algebra. This splits into an inner and an outer part

(8)$1 \to \mathrm{Inn}(A) \to \mathrm{Aut}(A) \to \mathrm{Out}(A) \to 1 \,.$

By comparison with the case of ordinary geometries, we address the outer part as the group of diffeomorphisms of our target space. The inner part, however, is identified with the group of gauge transformations.

Next we need a Hilbert space on which this algebra $A$ is represented by compact operators. We take

(9)$H = H_\mathrm{ext}\otimes H_\mathrm{int} \,,$

where

(10)$H_\mathrm{ext}$

is the Hilbert space of square integrable sections of a spinor bundle on our 4-manifold $X$ and

(11)$H_\mathrm{int}$

is a certain representation of the algebra $A_\mathrm{int}$ which is spanned by a basis which contains precisely the list of elementary fermions that appear in the standard model.

So this basis consists of

• the quarks $\array{ u_R & u_L \\ d_R & d_L }$ each of which comes in three colors;
• and the leptons $\array{ & \nu_L \\ e_R & e_L }$

and the antiparticles to each of these… and two more copies of all of this, giving the three generations of fermions.

All-in-all this is a list of

(12)$((4\times 3 + 3)\times 2) \times 3 = 90$

particles (three generations of particles and and antiparticles, each consisting of 4 quarks in three colors, a neutrino and a left- and a right-handed part of the electron) and our Hilbert space is simply the span of these 90 basis elements (everything here is over $\mathbb{C}$)

(13)$H_\mathrm{int} \simeq \mathbb{C}^90 \,.$

Next, we need a representation of $A$ on our Hilbert space $H$. The rep of $A_\mathrm{ext}$ on $H_\mathrm{ext}$ is the obvious one, obtained by multiplying spinors with complex numbers.

The rep of $A_\mathrm{int}$ on $H_\mathrm{int}$, encodes crucial information about the particle physics we want to obtain.

Writing down this rep in detail is not particularly enlightning. In his latest paper, however, Connes explains (in prop. 2.2) that we can understand this representation naturally as the direct sum of all inequivalent irreducible odd $A_\mathrm{int}$-bimodules.

Next, we need to specify the grading $\gamma$ and the real structure $J$ on our Hilbert space $H$.

This involves a choice of a couple of signs, and it is just this choice of signs which the recent progress consists of. With

(14)$\gamma = \gamma_\mathrm{ext}\otimes \gamma_\mathrm{int}$

and

(15)$J = J_\mathrm{ext} \otimes J_\mathrm{ext}$

we take the external part to be given by the standard choice

(16)$\gamma_\mathrm{ext} = \gamma^5$

and

(17)$J = \mathrm{Id}$

(hm, I think - I need to check that, corrections are appreciated).

Writing down the components of $\gamma_\mathrm{int}$ and $J_\mathrm{int}$ is again not really illuminating (though $J$ can essentially be thought of as charge conjugation), but, as with the rep of $A_\mathrm{int}$, Connes shows that there is natural way to understand both in terms of the isomorphism of the above mentioned bimodule with its contragradient one.

Finally we need of course to specify our Dirac operator

(18)$D = D_\mathrm{ext}\otimes \mathrm{Id} + \gamma_1 \otimes D_\mathrm{int} \,.$

As described before, we take

(19)$D_\mathrm{ext} = D_0 + \delta D$

to be the ordinary Dirac operator $D_0$ acting on sections of the given spinor bundle on our Riemannian 4-manifold $X$ plus a part $\delta D$ containing what for ordinary geometries is called the minimal coupling of fermions to gauge forces.

Since we think of $D$ as encoding a metric, we may think of $\delta D$ as “internal parts” of the metric, as in Kaluza-Klein theory.

One can derive the form of this part nicely in terms of inner automorphisms of the algebra $A$, and it turns out to be of the form

(20)$\delta D = A + J A J^\dagger \,,$

where $A = \sum_i a_i [D,a_i]$ for $a_i \in I$.

For ordinary geometries, these terms evidently look like the usual minimal coupling terms $\sum_i a_i [\gamma^\mu \nabla_\mu, a_i] = \gamma^\mu A_\mu$ to the gauge bosons.

The crucial point here is that we also get a contribution from the internal metric

(21)$[D_\mathrm{int}, a]$

and this gets interpreted as the Higgs boson.

The internal Dirac operator $D_\mathrm{int}$ can, since it acts on a finite dimensional space, simply be thought of as a large matrix, satisfying a couple of conditions. After computing the spectral action (but we need just the obvious term $\langle \psi , D \psi \rangle$ to see that) it will turn out that this matrix is essentially the matrix $Y$ of Yukawa couplings. So we read off all these couplings from our favorite particle accelerator and write them as entries into our internal Dirac operator.

That’s it. A spectral triple.

Maybe to summarize, here is what the main ingredients of the standard model correspond to in the spectral description:

• gauge group $\leftrightarrow$ inner automorphisms of algebra $A$
• diffeomorphism group $\leftrightarrow$ outer automorphisms of algebra $A$
• fermion species $\leftrightarrow$ basis elements of $H_\mathrm{int}$
• number of generations $\leftrightarrow$ choice of size of $H_\mathrm{int}$
• gauge bosons $\leftrightarrow$ external/internal part of the metric, as in Kaluza-Klein
• Higgs bosons $\leftrightarrow$ another external/internal part of the metric
• gauge couplings $\leftrightarrow$ encoded in external/internal part of the metric
• Yukawa couplings (fermion masses) $\leftrightarrow$ encoded in the internal metric

The next step is to analyze this spectral triple. To compute the dimensions of the space it describes and to determine the spectral action it induces.

Posted at September 8, 2006 12:34 PM UTC

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### history

A very complete listing, partly clickable, of the articles and eprints related to the first ten years of this model can be found in my webpage ncactors.html

Posted by: Alejandro Rivero on September 8, 2006 6:08 PM | Permalink | Reply to this

### … and two more copies

Perhaps the most unsatisfying point of the new model is that generations are still here by hand. That means that CP violation is not a requisite, in the sense that a one-generation standard model does not have CP violating phases but is consistent with the requisites of Connes his current model.

Here at the dinner tables, as in the internet, there was some buzz about if this 6 mod 8 was really 6 or 22, so really the total geometry can be either 10 or 26. I will enjoy if we could use somehow the generation number to count the “number of rounds” we have gone around the KO dimension clock (so 6 for one, 6+8 for two, 6+8+8=22 for three).

Incidentally, there is a minor NCG based argument to select d=26, and it is that when one uses the GVF normalisation of the Dixmier trace then the proportionality coefficient between this trace and the Einstein Hilbert action becomes (d-2)/24 (physics/0409022) This normalisation is very natural, and for instance it used by Martinetti in his PhD thesis.

Posted by: Alejandro Rivero on September 8, 2006 6:23 PM | Permalink | Reply to this

### reduction and oxidation

Hi Alejandro,

[…] count the “number of rounds”

I have a certain physics intuituion (possibly wrong - but possibly right) which makes me expect that something along the following lines can be made precise. Maybe you have sees something like this addressed anywhere:

There should be something like a moduli space of all spectral triples (maybe of all sufficiently well-behaved spectral triples).

I expect that there are certain quantities we could associate to a given spectral triple such that they are invariant on connected components of this moduli space.

In particular, I am imagining that it should be possible to make the idea of a spectral triple being a “compactification” precise by

- finding a continuous path through the moduli space of spectral triples that starts at a “classical” geometry (with a commutative algebra) and ends at the given “compactification” geometry, which may be non-commutative.

It might be that a quantity like the KO-dimension is in fact an invariant along such paths.

Could something like this be true?

If so, we could determine the “number of rounds” around $\mathbb{Z}_8$ that the KO-dimension goes through by decompactifying our standard model spectral triple. So we would try to see if we can continuously deform it and flow through the space of all spectral triples to a classical one that is the oxidation of our noncommutative one.

For classical geometries classical dimension mod 8 coincides with KO-dimension. So it would then make sense to say that the dimension of the geometry of our spectral triple (of which we see only the KO-shadow) is that of the classical geometry oxidizing it.

Could anything like this work? Is there any literature that might be related?

Posted by: urs on September 9, 2006 2:09 PM | Permalink | Reply to this

### Re: reduction and oxidation

bq. all sufficiently well-behaved spectral triples

Krajewski took pains to isolate, or classify, the discrete spectral triples, meaning these having both spectral and KO dimension equal 0. Now it seems that this work should be renovated, or the problem atacked from other point of view.

finding a continuous path through the … space of spectral triples …

The concepts of “isospectral deformation” and “Morita equivalence” could be of some use here. And yes, probably the KO dimension would be sort of invariant here. But I do not see how “decompactification” could be implemented (I would call it “de-noncommutativisation” as the resulting decompactified space could still be a compact one).

Another idea could be to see if we can start from a one-generation spectral triple and tensor it times a zero dimensional (thus 8 dimensional) to jump to a two-generations triple. I raised this topic in the Chinese pub yesterday; the main problem is that, depending on the spectral triple we use to multiply with, we increase the symmetry groups and we need further unifications and/or symmetry breakings. On the other hand I would be happy having three Higgs, as Koide has done some interesting work on three-Higgs models as a basis of his formula.

Could anything like this work? Is there any literature that might be related?

Guess that anybody (anybody interested, this is) is taking the weekend to see if anything like this could work. We would need some tool beyond KO theory, something like the the distinction between topological and algebraic K-theories (one gets Bott periodicity, another “spirals up” across this periodiciy); I am no paranoid with this, but coming to the institute this evening I noticed a blackboard with sort of a lecture on K theory, Hochschild cohomology and cyclic cohomology.

Literature? Standard Atiyah et family, any random book in K theory. KO is also quickly introduced in the book on “spin geometry” (from Michelson???).

Posted by: Alejandro Rivero on September 9, 2006 4:17 PM | Permalink | Reply to this

### Re: reduction and oxidation

The concepts of “isospectral deformation” and “morita equivalence” could be of some use here.

[…]

Literature? Standard […]

Very true, right. Essentially we are doing K-homology. I should think Fredholm modules instead of spectral triples. Of course.

But I do not see how “decompactification” could be implemented (I would call it “de-noncommutativisation” as the resulting decompactified space could still be a compact one).

Right. I was using “compactified” in the sloppy physics sense of “sent to small volume”.

Let’s check if the standard model spectral triple is smoothly connected to one for which metric dimension mod 8 coincides with KO-dimension - or, if possible, one which is even commutative.

Maybe I am way wrong. But the formal high energy physicist in me suspects that most of these noncommutative spectral triples should be connectable by a continuous path in some sensible moduli space to something commutative.

Posted by: urs on September 9, 2006 8:04 PM | Permalink | Reply to this

### Re: reduction and oxidation

I should think Fredholm modules instead of spectral triples. Of course.

Yep, this is even more true now that the spectral dimension from the finite Dirac operator doesnt matter. In Connes book, the Fredholm module is usually extracted by removing any metric information from the Dirac operator.

Does Baez has some of these little introductions he knows how to do, but about K-theory? It is a notationally awfull topic, with all these variants of algebraic, topological, real, Real, etc… Nor to speak of CC, HC, HH and their variants.

the formal high energy physicist in me suspects that most of these noncommutative spectral triples should be connectable by a continuous path in some sensible moduli space to something commutative

This has been my first impression too. But note that we already got a lot of pain trying to use series of finite spectral triples to approach commutative spaces. Goeckeler-Schuecker non-go result for lattices was intended to be definitive, and in fact any further effort in the lattice way was discouraged; on the other side I believe than some success was got on q-spheres.

Posted by: Alejandro Rivero on September 10, 2006 1:06 PM | Permalink | Reply to this

### Re: reduction and oxidation

Alejandro Rivero writes:

Does Baez has some of these little introductions he knows how to do, but about K-theory?

Not really. In week149 I explained how to get “generalized cohomology theories” like the various flavors of topological K-theory from gadgets called “infinite loop spaces”. Near the end of week199 I explained how to get infinite loop spaces from symmetric monoidal categories. If you stick these together, you’ll get a taste of how homotopy theorists think about topological K-theory. However, to understand the K-groups $K^i$ for $i$ negative from this perspective, one really needs full-fledged spectra, not just infinite loop spaces.

There’s also a lot of less fancy stuff one should learn, like the basic definitions and properties of various flavors of topological K-theory, up to and including Bott periodicity. I’ve found Karoubi’s book to be quite nice for this, so you might like this:

and then this:

• Max Karoubi, K-theory: an Introduction, Springer, Berlin, 1978.

For K-theory and the Atiyah-Singer index theorem, I recommend Atiyah’s book K-theory and the Seminar on the Atiyah-Singer Index Theorem edited by Richard Palais.

I find the new Springer-Verlag Handbook of K-theory to be somewhat obnoxious, because while enormous, it somehow doesn’t have room for a simple explanation of the basic definitions and facts about topological K-theory.

I enjoyed this:

• B. Blackadar, K-Theory for Operator Algebras, Math. Sci. Res. Inst. Publ. No. 5, Springer, Berlin, 1986.

But, I think this book assumes you’re fond of C*-algebras and Banach algebras.

Posted by: John Baez on September 11, 2006 5:29 AM | Permalink | Reply to this

### Re: reduction and oxidation

Does Baez has some of these little introductions he knows how to do, but about K-theory?

I once tried myself, here. And then this discussion is supposed to emphasize the particular point of view we might desire here.

But I am not at all expert enough to sit down, without investing plenty of time, and compute isospectral deformations of the Connes standard model spectral triple.

Posted by: urs on September 11, 2006 11:48 AM | Permalink | Reply to this

### Re: reduction and oxidation

Urs wrote:

But I am not at all expert enough to sit down, without investing plenty of time, and compute isospectral deformations of the Connes standard model spectral triple.

I don’t know much about “isospectral deformations”, but one thing I know is that the algebra

(1)$A_\mathrm{int} = \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$

is rigid. In other words, if we change the structure constants of this algebra a tiny bit while keeping an associative algebra, we get an isomorphic algebra. That’s because this algebra is semisimple.

I would like to go further and change “a tiny bit” to “in a continuously varying way”, but I’m unsure of my reasoning here.

Anyway, I’m curious about what sort of deformation of the whole spectral triple would actually be interesting here - and have a chance of being nontrivial.

A related question:

Are there “nongeometric phases” of string theory where the internal space is a noncommutative algebra that’s able, by some sort of “tuning of parameters”, to shrink down to a tiny algebra like this $A_{int}$?

Posted by: John Baez on September 11, 2006 5:08 PM | Permalink | Reply to this

### Re: reduction and oxidation

the algebra $A$ is rigid.

Okay. So we probably cannot expect it to be continuously connected to anything commutative directly.

Isospectral deformations might however yield much more freedom for deformations.

For instance, one thing that comes to mind is applying algebraic T-duality.

In the imagery of my working hypotheses, this would make good sense, since it should correspond to passing from a space of vanishing size to a space of infinite size, while keeping the observable physics the same.

For this to work (in its simplest form), we’d need an action of $\mathbb{R}^6$ on the Connes algebra $A$. Given such an action, the T-dual algebra would be the semidirect product of $C^*$ algebras

(1)$A \rtimes \mathbb{R}^n \,.$

While probably not directly relevant, hep-th/9904064 by Szabo and Lizzi describes a general approach for understanding such dualities in terms of spectral triples.

Are there “nongeometric phases” of string theory where the internal space is a noncommutative algebra that’s able, by some sort of “tuning of parameters”, to shrink down to a tiny algebra like this $A_\mathrm{int}$?

One would have to extract such algebras from Gepner models, for instance, as some (point particle) limit of the algebra of vertex operators, I guess.

But, off the top of my head, I don’t recall if I have seen such a computation.

Posted by: urs on September 11, 2006 6:16 PM | Permalink | Reply to this

### Re: reduction and oxidation

We got some experts on non-geometric phases during the week, and nobody stressed such posibility.

Posted by: Alejandro Rivero on September 12, 2006 9:29 AM | Permalink | Reply to this

### generations

Another idea could be to see if we can start from a one-generatio spectral triple and tensor it times a zero dimensional (thus 8 dimensional) to jump to a two-generations triple.

Do I understand correctly that you are suggesting that for each internal 8 dimensions we get one additional generation of fermions?

I have no intuition for that relation. What are the indications that this might be true?

Posted by: urs on September 9, 2006 8:10 PM | Permalink | Reply to this

### Re: generations

None except that now it is the only quantity we put by completely by hand in the model so it seems sensible to keep an eye on it.

In the primitive ConnesLott family of models (or perhaps in Coquereaux et al. models?) there was an argument for more than one generation, related to get a non zero copling for the Higgs; this possibility dissapeared time ago. Then there was an argument about needing 3 generations in the old Connes-Chamseddine model and action because the only way out seemed to be to keep one neutrino massless (so Poincare duality was preserved) and to give mass the other two. Now in this construction (Connes-Barrett?) we can give mass to neutrinos in any number of generations, so the previous requeriment has dissapeared too. The only extant argument for three generations could be to argue philosophically for the need of a violation of CP (thus of T).

A first idea, the one we discarded in the pub, was to look for a mechanism multiplying the spectral triple times a 8-dimensional finite one and then somehow collapse the new hilbert space of fermions so that is is only noticeable in the Cabibbo-K-M matrix. This naive product seems not able to work, because two successive products would get really a lot of extra vectors in the Hilbert space, plus unwanted groups coming from tensor product of the algebras of each spectral triple. In any case, any pausible argument working in the turn from one to two is likely to fail from two to three because it should not to duplicate the size of the Hilbert space, but only to add a copy of the original one.

The only argument “in pro” of a proliferation of algebras is that we could really need more than a minimal Higgs.

Posted by: Alejandro Rivero on September 10, 2006 12:37 PM | Permalink | Reply to this

### Re: generations

Now, I was thinking that if you tell a string theorist that “I am looking for a zero dimensional object that lives in dimension 2 mod 8” he will answer fast: you are looking for a Lorentzian unimodular even lattice. And if he is network-aware, he will redirect me to week95 (if not, I will probably get a rambling about Gross et al. 1985 PRL and the lattices $\Gamma_8, \Gamma_{16}$).

So, one would need to justify that each replication of the finite hilbert space is related to this periodicity in the dimension of the lattice, and surely to explain if/how the periodicity of lorentzian lattices relates to Bott or Clifford algebra periodicites (the later could be more feasible, but I havent see any paper on it neither). Third, the failure to find r.v=cte for lattices beyond Leech (ie beyond Lorentzian 25,1) would explain why the replication stops at three generations.

It could be interesting to find in the bibliography interpretations of the heterotic string where each generation spans 8 of the 24 transversal directions of the string. No idea if they exist.

The main caveat here is that Bott/Clifford 8-periodicity seems richer than the unimodular lattice periodicities.

Posted by: Alejandro Rivero on September 22, 2006 3:51 PM | Permalink | Reply to this

### Re: generations

each generation spans 8 of the 24 transversal directions of the string

But it has 24 transversal dimensions only when compactified to a $1+1$ dimensional spacetime! That’s two spatial dimensions short of what is observed.

I see that you would like to find out if there could be any relation between the number 3 in

(1)$3 \times 8 = 24$

and the number of observed generations of fermions.

But apart from the fact that it is a 3 in both cases, I cannot quite see yet what kind of relation that might be - at least not in the context of the heterotic string.

But let’s forget the heterotic string for a moment, and just concentrate on the Connes model.

In that model, the number of generations is essentially determined by the choice of representation of the internal algebra on the internal Hilbert space.

Now, the real structure and the grading on that Hilbert space is what determines the indernal KO-dimension.

You would like to see if both are related through the magic of even unimodular lattices.

So I guess the question is then: is there naturally associated a noncommutative algebra with representation by bounded operators to any even unimodular lattice, an algebra that would encode something like the noncommutative geometry of the lattice?

Hm. Is there?

Posted by: urs on September 22, 2006 4:31 PM | Permalink | Reply to this

### Re: generations

Er, just a minor correction: a string has 24 transversal directions when it lives in 26 dim space time (and 8 when it lives in 10 dim space time). But I agree that there is no clue about 24=8*3, I was just hoping that some papers between the huge bibliography of string theory could speculate on this.

I guess the question is then: is there naturally associated a noncommutative algebra with representation by bounded operators to any even unimodular lattice, an algebra that would encode something like the noncommutative geometry of the lattice?

I guess this is the question, yes. I’d add that some corolary about Bott periodicity in the lattices would be welcome.

Posted by: Alejandro Rivero on September 22, 2006 5:03 PM | Permalink | Reply to this

### Re: generations

a string has 24 transversal directions when it lives in 26 dim space

Okay, I see. I thought that’s not the “transversal” we were talking about, but that we were talking about the number of dimensions “transversal” to the uncompactified observed ones.

Otherwise your argument would be mixing up properties of the internal space with that of the “external” one, it seems.

Posted by: urs on September 22, 2006 5:14 PM | Permalink | Reply to this

### Re: generations

Ah, well, I see, the point is that in Connes models we multiply the 4 dimensional space time times the 0 dimensional (but dimension 6 in K0 terms) one, so in some sense the later space is the compactified one. But note that this 0/6 dimensional space does not use a hilbert space related straightforwardly to these dimensions; it uses instead the fermions of the standard model.

Thinking on some kind of analogous discrepance in strings, I remembered of trasversal directions. They live in dimension 26, compactify to 4+22, but vibrate in 24 transversal directions. So 24 is the real issue. In fact I had already used this concept two years ago in a different idea, so it was just there in my brain inside some neuronal memory trap :-) . All the Kaluza Klein modes in string theory is just headache.

The advantage of Connes is that there are not Kaluza Klein remmants. The spectral dimension is 4 dimensional, the KOtheoretical is 26.

The main worry I have now is that the 26 of lattices (and of strings) is Minkowskian, while the 26 of Connes is Euclidean.

Posted by: Alejandro Rivero on September 22, 2006 6:10 PM | Permalink | Reply to this

### Re: generations

we multiply the 4 dimensional space time times the 0 dimensional (but dimension 6 in K0 terms) one, so in some sense the later space is the compactified one.

Yes.

But note that this 0/6 dimensional space does not use a hilbert space related straightforwardly to these dimensions; it uses instead the fermions of the standard model.

Yes. This is exactly what makes your idea sort of interesting. The number of generations of fermions is directly related to the dimension of the internal Hilbert space in Connes’ model: the dimension of the Hilbert space is a fixed number times the number of generations.

And intuitively we expect this Hilbert space to be the larger the higher the dimension of the geometry it describes is.

For instance, let the geometry be described by the space of complex-valued functions on an elementary $n$-cube in a $\mathbb{Z}^n$-lattice. Then the corresponding Hilbert space (of the canonical representation) has dimension $2^n$.

(This is of course just a plausibility argument, in no way anything else.)

Posted by: urs on September 23, 2006 1:41 PM | Permalink | Reply to this

### Re: generations

Yep, the idea seems interesting but the more I think about it, the more I am afraid of! It could be reinterpreted to say that uncompatified string theory is right and the “only” problem is that they did not understand how to attach the background. The recipe should be: separate the 24 transversal and 2 “lightconed” degrees of freedom, and attach 4dim space time only to the latter.

The scheme does nothing to justify the hierarchical texture of generations. This should be my main point against trying a formulation along these lines.

Besides, another minor problem I have is that there are no 24 but only 12 different mass eigenvalus in the fermions, so one must argue for Weyl fermions to duplicate them. This is only a minor issue, but disturbing.

Posted by: Alejandro Rivero on September 24, 2006 4:21 PM | Permalink | Reply to this

### Re: generations

I am doing a short flight to Zaragoza for some paperwork; while waiting at Stansted I got a conjecture over this conjecture: perhaps we are seeing not the Leech lattice, but the quotient of the Leech lattice by its dual. This is -being the lattice selfdual- naively a point, but NCG is very well known to be able to preserve information about quotients. Why this conjecture? Because in lattices, the 8-periodicity is claimed -I have been told- for this kind of quotients. This is usually stated as Milgram’s theorem in a few papers of Conway’s school. A sort of recitprocity theorem or corolary.

Posted by: Alejandro Rivero on September 27, 2006 5:22 PM | Permalink | Reply to this

### Re: generations

Hi Alejandro,

I am doing a short flight to Zaragoza

I wish you a nice trip!

conjecture over this conjecture: […]

This all sounds interesting. But it remains pretty vague. I feel that one would probably see clearer if there were a way to canonically associate certain finite-dimensional non-commutative algebras to all these lattices.

If you run into anyone who might know about the existence or non-existence of such algebras, you should ask.

I am not expert enough on this stuff to know. But probably some octonion will play a role here and there.

Or, asking the other way around, is there any way Connes’ internal algebra can be related to anything octonionic, maybe embedded into it or something?

Posted by: urs on September 27, 2006 6:00 PM | Permalink | Reply to this

### 24

Interesting. I vaguely recall having seen that before. But I forgot about it. You write

the right normalised form of the fundamental theorem for commutative spectral triples has evolved to show explicitly the extra factor […]

(1)$S(D) = -\frac{n-2}{24} \int_M \,s \, \sqrt{\mathrm{det}g} d^n x$

concluding:

Thus we claim that a perturbative quantisation of commutative spectral triples gives gravity only if the dimension of the triple is $n=26$.

So you are saying the right place to read more on the “correct form of the pre-factor” is math-ph/0112038?

That’s pretty interesting…

By the way, you also write

[…] or commutative spectral triples can be obtained as a limit of bosonic strings.

I know I have said the following now a couple of times already, but it deserves maybe to be said once again.

Chamseddine claimed that he has checked to first nontrivial order that using the superstring’s Dirac-Ramond operator in the spectral action does reproduce the familiar string effective background field action.

Posted by: urs on September 9, 2006 2:33 PM | Permalink | Reply to this

### Re: 24

No, the right side to read about it is GraciaBondia-Varilly-Figueroa book; but it is not online except as an scan in P2P networks, so a fast view of the formula can be found in Martinetti’s math-ph/0112038.

My own note, I am not proud of it. As you can see it was even redirected to physics/ as being overly (and surely, in the last pages, wrongly) speculative.

Generically it is not a surprising coincidence. As I hear from a PI researcher past yesterday at St John, “there are not many two-dimensional objects” in differential geometry.

Posted by: Alejandro Rivero on September 9, 2006 3:02 PM | Permalink | Reply to this

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

In case anyone here is trying to read Connes’ new paper, here are a couple of typos he mentioned when I expressed my puzzlement over a certain issue.

In Proposition 2.2, it should read $\mathcal{M}_F$ instead of $\mathcal{H}_F$ (this is how one gets the representation of the algebra: as the direct sum of all odd irreducible ones).

In Definition 2.6, it should read $(\lambda, \lambda, 0)$.

Posted by: John Baez on September 9, 2006 3:03 AM | Permalink | Reply to this

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

There is also some issue about if mixing of quarks and leptons can happen, it seems that Barrett and Connes discussed about it and concluded it doesn’t; perhaps the issue will be commented next Monday in Barrett his lecture.

Posted by: Alejandro Rivero on September 9, 2006 2:55 PM | Permalink | Reply to this

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

[…] next Monday in Barrett’s lecture.

If you hear any news, please let us know.

Posted by: urs on September 9, 2006 8:14 PM | Permalink | Reply to this

### Background independence, boundedness

Two things about the philosophy of NCG have bugged me for a long time.

The first is background independence. AFAIU, a spectral triple (A,H,D), A commutative, is equivalent to a manifold equipped with a metric structure. Thus I worry that Connes is sneaking in a background metric in the guise of the Dirac operator. Perhaps this is not so, but at least the metric is treated differently from the other fields since it is encoded in the Dirac operator.

A spectral double (A,H) would encode background independent data. Unfortutately, then we lose too much structure; not just the metric but also the smooth structure, which is assumed as a background in GR. Data which would encode a smooth but not metric structure would be (W,H), where W is the Witt algebra of vector fields acting on H.

My second worry is that A is supposed to be represented as bounded operators on H. This is fine in QM, of course, but would you really encounter infinities in QFT (renormalization, anomalies,…) if all operators were bounded?

### Re: Background independence, boundedness

I have tried to emphasize the analogy in all four postings.

You have a single particle, described by a single Dirac operator. That operator describes propagation of that particle on a fixed background. The operator is that background.

But the dynamics of the particle induces a dynamics for the background. In Connes approach this is so by fiat, since he defines the dynamics of the background to be determined by the spectral action.

So with the spectral action providing an “effective field theory on target space”, we may now vary the background fields - hence vary the Dirac operator - and weight by the spectral action.

It’s precisely analogous to what happens in perturbative string theory, if you think about it.

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

### Re: Background independence, boundedness

My second worry is that A is supposed to be represented as bounded operators on H. This is fine in QM, of course, but would you really encounter infinities in QFT (renormalization, anomalies,…) if all operators were bounded?

Yes, and in fact the most rigorous approaches to quantum field theory work with C*-algebra of bounded operators rather than unbounded ones. A unbounded self-adjoint operator $A$ can be recovered from the 1-parameter group of bounded operators $\mathrm{exp}(i t A)$, and the exponentiated form allows one to state relations like the canonical commutation relations in a more powerful manner (the Weyl relations, or more generally Lie group rather than Lie algebra relations). The infinities of quantum field theory get repackaged in the fact that a given C*-algebra typically has unitarily inequivalent representations.

So, this is no problem if one is careful.

Your question about the background-independence of noncommutative geometry seems more interesting to me, meaning that I don’t feel sure of the answer.

Posted by: John Baez on September 11, 2006 4:40 PM | Permalink | Reply to this

### Re: Background independence, boundedness

My second worry is that A is supposed to be represented as bounded operators on H. This is fine in QM, of course, but would you really encounter infinities in QFT (renormalization, anomalies,…) if all operators were bounded?

We need to be careful not to confuse operators of the quantum theory of Connes’ superparticle with operators of the target space field theory induced by that.

$A$ is an operator on the Hilbert space of that single particle (for fixed background, by the way), and we certainly want it to be bounded. It is supposed to be just given by multiplication and Clifford multiplication.

In the target space theory defined by the spectral action, $A$ will become quantized again (second quantization) and whether or not this second quantized field operator is bounded or not is a rather different question.

Your question about the background-independence of noncommutative geometry seems more interesting to me, meaning that I don’t feel sure of the answer.

The spectral action is a functional that eats a Dirac operator $D$ and a spinor $\psi$ living in a Hilbert space $H$ of a single “superparticle” and spits out an amplitude for them.

So whatever is encoded in $D$ and $\psi$ is not background, while any other structure that is present is background.

I keep emphasizing that it may be helpful to compare this to what happens to 2-spectral triples otherwise known as 2D SCFTs.

Posted by: urs on September 11, 2006 5:29 PM | Permalink | Reply to this

### Re: Background independence, boundedness

So whatever is encoded in $D$ and $\psi$ is not background, while any other structure that is present is background.

For completeness, I should add that we usually want to fix part of the data inside $D$ as fixed background.

In Connes’ formulation of the standard model, we actually want to keep the metric on the compact space, namely the operator $D_\mathrm{int}$ fixed, and only vary the metric on the remaining 4D space (which yields 4D gravity) as well as the “internal fluctuations” of the metric, as Connes calles them, which are really the external/internal parts of the metric that yield, by Kaluza-Klein logic, the gauge fields (and the Higgs).

So of $D = D_\mathrm{ext} + \gamma^1 \otimes D_\mathrm{int}$ we only want to vary $D_\mathrm{ext} = D_0 + \delta D$ (in my notation).

Notice that the part we don’t vary determines the dimensionless parameters of the standard model.

If we wanted to, we could justifiably refer to this as picking a point in the landscape of $(d_\mathrm{KO} = 4+6)$-compactifications of spectral triples.

But maybe we don’t want to.

Posted by: urs on September 11, 2006 6:50 PM | Permalink | Reply to this

### Why should it work?

Now I’m confused. Does the Connes model contain gravity? If so, where is the Einstein action? If not, there is no way that it could be, or should be, background independent. But then there should be a way to couple it to gravity.

I have another question, which should be asked of every putative theory of quantum gravity: Why should it work? We know that ordinary QFT is incompatible with gravity, and thus QG must modify either QFT or gravity or both. What is the physical reason why non-commutativity should cure these problems?

And yes, I can answer the corresponding question, see math-ph/0603024, p 9.

### Re: Why should it work?

Does the Connes model contain gravity? If so, where is the Einstein action?

The Einstein action is among the first terms of the expansion of $\mathrm{Tr}(f(D))$.

should be asked of every putative theory of quantum gravity

Connes approach is not a theory of quantum gravity. It is instead a way to encode the action functionals for Einstein gravity and the standard model.

and thus QG must modify either QFT or gravity or both.

Yes. That’s why Connes’ spectral triple can only be a low-energy approximation to the real thing. He does discuss possible completions of his action in the very last section of his latest paper.

What is the physical reason why non-commutativity should cure these problems?

Notice that at the present stage 4D spacetime is assumed to be perfectly commutative in this work. “The noncommutativity” of the specific model we are discussing is not there to cure any problems, but to elucidate the structure of the standard model.

To summarize, what Connes’ standard model spectral triple shows is this:

There is a comparatively simple choice of a spectral triple $(A,H,D)$ such that the associated spectral action is, to first order, the Einstein-Hilbert action coupled to the action of the standard model.

Posted by: urs on September 12, 2006 9:39 AM | Permalink | Reply to this

### Re: Why should it work?

The Einstein action is among the first terms of the expansion of Tr(f(D)).

Ah - I didn’t realize this, but thought that it might just be a Yang-Mills term. OK, so Connes’ model does contain gravity.

Yes. That’s why Connes’ spectral triple can only be a low-energy approximation to the real thing. He does discuss possible completions of his action in the very last section of his latest paper.

Do you mean section 6, Interpretation? I found that vague, to say the least.

“The noncommutativity” of the specific model we are discussing is not there to cure any problems, but to elucidate the structure of the standard model.

Hm, hm. Still, it might be worthwhile to emphasize that if Connes model is essentially a field theory, then it should encounter the usual problems with gravity, unless there is some reason otherwise.

### Re: Why should it work?

Ah - I didn’t realize this, but thought that it might just be a Yang-Mills term. OK, so Connes’ model does contain gravity.

Yes. The achievement here, too, is in a way cosmetic, since the nice thing is how both the Yang-Mills action as well as the Einstein-Hilbert actions (as well as their coupling) drop out of the single expression $\mathrm{Tr}(f(D))$.

It as a form of a generalized Kaluza-Klein mechanism.

Do you mean section 6, Interpretation?

Yes.

I found that vague, to say the least.

Right, sure. It just mentions some speculations.

Hm, hm. Still, it might be worthwhile to emphasize that if Connes model is essentially a field theory, then it should encounter the usual problems with gravity, unless there is some reason otherwise.

Yes, exactly.

So Connes’ model makes no direct progress in understanding quantum gravity or “physics beyond the standard model”.

His achievement is to point out that the experimentally known structure of gravity coupled to the standard model can very elegantly be encoded in a much more concise structure.

He makes this point very nicely in his paper by first writing down his spectral triple and the spectral action (a couple of lines) and then spelling out the full standard model action, which takes a full page, single-spaced.

So Connes’ formulation can be relevant to quantum gravity only in so far as a more elegant description of what is known might make it easier to guess what is not known.

You are all welcome to see if Connes’ formulation of the standard model suggests anything about your favorite approach to quantum gravity.

Personally, I noticed that one rather natural attempt to “complete” Connes’ description to a theory going beyond the standard model might be to realize the spectral triples as limits of 2-spectral triples.

But that’s just me.

Posted by: urs on September 12, 2006 12:32 PM | Permalink | Reply to this

### Connes conjectures

Today ( Monday 9th september) has been an intermediate session, the end of the trilogy being to be past tomorrow (and perhaps at the same time the Connes-Marcolli-Chammseddine paper will be released). Well, the point is that Alain has taken some time to do remarks about his own guesses. One of them is a old one: that the algebra comes from a q-deformation of SU(2)xSU(2), because it looks very much as the truncation up to spin one of the series (M_1 x M_2 x M_3 x M_4 x…) of matrices over C

The other remark, and it is news to me, is that the number of generations could be related to the fundamental group of the non commutative manifold: the NC Manifold of the standard model could be not simply connected, and the standard tools of K-theory fail to register it. We should be getting some cross product of the algebra of the manifold times the fundamental group; if the fundamental group is abelian that should be various copies of the same spectral triple except for the dirac operator which should vary. Or say otherwise, “downstairs” we have one generation and we lift Dirac operator to the universal cover of the manifold and we see that we need various copies.

I do not claim to be understanding, just taking down a fast note, because I do not know if this remark will appear in the paper. I have preferred not to disturb Alain asking about it; he was busy talking with Barrett after the lecture.

Posted by: Alejandro Rivero on October 9, 2006 5:13 PM | Permalink | Reply to this

### Re: Connes conjectures

the NC Manifold of the standard model could be not simply connected,

That’s interesting. This is the kind of information I was hoping for when I said it would be nice to understand the internal space as something like a limit of an ordinary space becoming tiny.

So apparently Connes says there is a way to associate a fundamental group to the “space” described by the algebra $\mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$?

How does one detect such a feature? I mean: how does one define the fundamental group associated to a given noncommutative algebra?

Posted by: urs on October 9, 2006 5:38 PM | Permalink | Reply to this

### Re: Connes conjectures

How does one detect such a feature. Nobody knows. Connes did the remark when he was going to do a (different?) remark about Poincare Duality. It seems he is thinking hard about the reshaping of the axioms.

I am afraid that if the fundamental group enters play, this is starting to look pretty stringy business. Parhaps I would post on this on my old abandoned blog (physcomments, scheduled to disappear at the end of 2007)

Sorry by the “x” in the previous post, it was to meant “+”, of course. This could be relevant because of a third remark: that the current model uses two copies of the quaternions, thus is more similar to M_2 matrices that the old one.

Posted by: Alejandro Rivero on October 10, 2006 4:42 PM | Permalink | Reply to this

### Re: Connes conjectures

Nobody knows.

Wait, maybe I misunderstood you.

I thought you said that

a) Connes knows a way to compute the “fundamental group” associated to a noncommutative algebra

b) Connes conjectures that the number of generations is related to the fundamental group associated this way to $A = \mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$.

Now I am not sure about a). Is it that Connes says he knows how to the define the fundamental group of an algebra, but hasn’t told anyone yet how to do this?

Posted by: urs on October 10, 2006 4:50 PM | Permalink | Reply to this

### Re: Connes conjectures

Indeed I only can assert b). And note that it is not only associated to A, but also the Dirac operator and the Hilbert space -the rest of the data of the spectral triple- are relevant in this situation.

As for a), yep, well, I agree that perhaps Connes knows, but he did not tell us :-) My guess about the remark he did not, the one on Poincare Duality, is that it was about how the new model forces Poincare Duality to be independiently asserted in quark and lepton sectors. At least this is the content of a “personal communication” to Marseille model builders last month.

(He started the excourse by telling that all the K stuff of Milnor, Quillen, etc, was doing some assumption of simply connecteness. Marcolli, from the public, remarked something other implicit assumptions, but I did not catch the remark)

Posted by: Alejandro Rivero on October 11, 2006 10:31 AM | Permalink | Reply to this

### Re: Connes conjectures

Thanks a lot for the all information, Alejandro!

And note that it is not only associated to $A$, but also the Dirac operator and the Hilbert space

Really? I thought the topology is entirely encoded in the algebra, and the fundamental group is a purely topological notion.

The Dirac operator and the Hilbert space should just add (geo)metric information to that.

(Like for instance the minimal length of a noncontractible path representing an element of the fundamental group.)

No?

Posted by: urs on October 11, 2006 10:46 AM | Permalink | Reply to this

### Re: Connes conjectures

I am not sure. In theory it is as you say, we start with A, and this is expected to fix at least all of the topology (or the Fredholm module thing) and then you add the dirac operator to get the geometry. But note that any number of generations can be imposed on this algebra, as the number of generations is really encoded in the Hilbert Space.

A funny thing is that beyond the multiplicity of generations, the representations of the standard model appear now very naturally, without needing to ask A to be a manifold (ie, to have Poincare duality). Of course a posteriori Poincare duality can be checked, and Alain says it still stands.

It seems Alain is taking a lot of worries about understanding the dangling point. Today he did a complete turn towards physics, remembering that there is no CP violation for less than three generations, and pointing out how it is related to a complex conjugation in the Dirac operator. But when asked if this was linked to past yesterday comment, the answer was a “I do not know”.

Well, next lecture will be 6th November (or 9th?). And sometime in the middle the paper of Connes-Marcolli-Chamseddine will appear (Marcolli was in the public today, perusing a heavily edited copy).

Posted by: Alejandro Rivero on October 11, 2006 2:35 PM | Permalink | Reply to this

### Re: Connes conjectures

But note that any number of generations can be imposed on this algebra, as the number of generations is really encoded in the Hilbert Space.

Okay, right. Probably what we need to answer such topological questions is the algebra $A$ and the representation

(1)$\rho : A \to B(H) \,,$

because the algebra which is really relevant is not so much $A$ itself, but rather the algebra $\mathrm{Im}(\rho) \subset B(H)$. I guess.

[…] appear now very naturally […]

What do you mean by saying “now”? Now that what has happened?

(It seems you are exposed to more first-hand information currently than can possibly be transmitted to a blog. :-) But it’s great to hear you report on Alain Connes’ most intimate thoughts at the moment.)

Well, next lecture will be 6th November (or 9th?).

If you happen to feel like writing more than just a couple of comments, we could possibly arrange a posting by you using our guest account.

Posted by: urs on October 11, 2006 2:49 PM | Permalink | Reply to this

### Re: Connes conjectures

Thanks! probably the guest account could be of more use, because math scripting has become partly disabled in physcomments and I do not feel I like to take pains to restore it. So yes, perhaps next week :-) Thanks

Meanwhile, to close here, a couple remarks:

- To specify some “topological” or scale-less properties, the F operator of the Fredholm module is relevant. Lets say that the differency between F and D is that the eigenvalues of D carry metric information too.

-By “now” I was referring to the last papers, but it is true I become aware only as Connes was doing the exposition and I even interrupted the lecture. The old point of view was explained by Florian Scheck. Remember than in physics, after you have imposed -also by hand- the standard model group, you must justify the representations of fermions, and this is argued from anomaly cancelation. Florian raised this point, and eventually it was related to some other condition in NCG, unimodularity. After the advent of Real spectral triples, I believed this condition had been subsumed into the Poincare duality axiom, but now it seems I was wrong: Poincare duality in the old models is forbidding the masive neutrinos, but does not speak about the anomaly. The spectral action principle, according 9605001, gives a “rationale” to justify unimodularity but it was still an ad-hoc requeriment.

Posted by: Alejandro Rivero on October 11, 2006 6:39 PM | Permalink | Reply to this
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