## September 6, 2006

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

#### Posted by Urs Schreiber

Alain Connes has a new report on recent progress in his old program of identifying the spectral geometry of the standard model coupled to gravity.

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

Similar results have simultaneously found in

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

In this first entry I’ll provide some background material. A followup will look at some of the details of the recent paper.

Recall from our discussion of the syntax of quantum mechanics that we can think of quantum particles, like those that appear in the standard model of particle physics, as being described by certain smooth functors

(1)$\text{SM}_\mathrm{WL} : 1\mathrm{Cob} \to \mathrm{Hilb} \,,$

where the domain is supposed to be some realization of the idea of the category of 1-dimensional Riemannian manifolds. (There is a sublety concerning the distinction between non-relativistic and relativistic QM, which, like many other subtleties, I shall ignore here. More discussion of this functorial way of looking at QM is going on here.)

My funny symbol $\text{SM}_\mathrm{WL}$ is short for standard model in worldline formulation. This formulation of quantum field theories like QED or the entire standard model, described for instance in

Christian Schubert
QED in the Worldline Formalism
hep-ph/0011331,

is particularly well suited for the ($n$-)categorical point of view on the world of particle physics.

A functor as above is specified by two pieces data:

• a Hilbert space $H$, which is the image of the single object $\bullet$ $H = \text{SM}_\mathrm{WL}(\bullet) \,,$ known as the space of states;
• an operator $\Delta$ on $H$, usually addressed as the Hamiltonian, such that $\text{SM}_\mathrm{WL}(\bullet \stackrel{t}{\to} \bullet) \;\;=\;\; H \stackrel{\exp(i t \Delta)}{\to} H \,.$

In the applications that we are concerned with in physics, there is also usually a third datum, namely

• a $C^*$-algebra $A$ represented by bounded operators on H, usually (vaguely) called (a sub-)algebra of observables

Back in the old days, Alain Connes noticed that this triple of data provided by quantum mechanics is a nice algebraic way to talk about Riemannian geometry.

To see this, notice that the nonrelativistic spinless boson propagating freely on a compact Riemannian manifold $X$ is described by a functor of the above sort such that

• $H = L^2(X)$;
• $\Delta$ is the Laplace operator on $X$ ;
• $A$ is the algebra of smooth (real/complex) functions on $X$ .

By analogy, any quantum system more complicated than the free spinless boson on a compact space can be regarded as defining a generalized notion of Riemannian geometry. Since the metric data is entirely encoded in the spectrum of the Hamiltonian, this approach is called spectral geometry.

(It is, slightly unfortunately, in fact often just addressed instead as noncommutative geometry.)

But in fact, both from the physic side as well as from the functional analytic side, we are lead to consider a slight refinement of this setup.

On the one hand, spinless bosons are rate in nature. In a way, spinning fermions are more “natural”.

On the other hand, Laplace operators are second order differential operators, hence not quite as elementary as first order operators, in a sense.

Both considerations lead us to the same conclusion.

Functorially, what happens is that we pass from the domain category of 1-dimensional Riemannian manifolds to (1,n)-dimensional super-Riemannian manifolds and pass from Hilbert spaces to graded Hilbert spaces.

(2)$\text{SM}_\mathrm{WL} : 1\mathrm{SCob} \to \mathrm{SHilb} \,.$

You can find a review of work by Stolz, Teichner & Markert on what this means in detail at the end of this entry.

It turns out that such functors are no longer characterized by a Hamiltonian but by a (generalized) Dirac operator $D$ on $H$, an odd-graded operator satisfying a few obvious algebraic conditions.

(3)$\text{SM}_\mathrm{WL}( \bullet \stackrel{(t,\theta)}{\to} \bullet) \;\; = \;\; H \stackrel{\exp(it D^2)(1 + i\theta D)}{\to} H \,.$

So our refined notion of a spectral triple $(H,D,A)$ involves a graded Hilbert space $H$, an operator $D$ of odd-degree and a representation on $H$ of the $C^*$-algebra $A$.

While it can be made plausible along the above lines why this notion of a spectral triple is useful, it is still amazing to me how very useful it is indeed.

It is hard to give a comprehensive idea of the available literature. Maybe I just point out the recent review

Alain Connes & Matilde Marcolli
A walk in the noncommutative garden
math.QA/0601054 .

On the other hand, while the noncommutative aspect of not-necessarily commutative spectral geometry has risen to immense popularity in the physics community, having given rise to the entire fields of noncommutative field theory (in field theory) - see for instance

Michael R. Douglas, Nikita A. Nekrasov
Noncommutative Field Theory
hep-th/0106048 -

and noncommutative D-brane configurations (in string theory), there is a remarkable scarcity of practitioners who take the spectral aspect seriously.

So far at least. Maybe Connes’ latest insights into the standard model help to change that.

Some notable exceptions from this rule that I am aware of are

• work on algebraic reformulations of central parts of string theory by Mathai Varghese and several others, mostly in the context of topological T-duality but more recently also, and more to our point here, addressing spectral reformulations of the nature of D-branes and RR-charges;
• work by Soibelman, Kontsevich, Roggenkamp, Wendland and others, which prominently involves spectral triples obtained from some sort of categorified version of what I was talking about above, namely the quantum mechanics not of point particles, but of 2-particles (= strings) as well as its “decategorification” obtained by taking the point particle limit.

The most farsighted application of these ideas to physics, however, has been followed by Connes and collaborators. Namely the idea of a spectral action principle.

It is known generally, that worldline theories of the kind I have discussed so far give rise to respective “effective” theories on target space ($\to$).

Connes proposed that, since all the information is encoded in the spectral triple, there must be a way to define that theory on target space (which, in phenomenologically viable applications, is nothing but the spacetime (parts of which) we observe) entirely in terms of natural operations on our spectral triple.

This idea is in fact well motivated by standard results obtained in heat kernel expansion

(4)$\mathrm{Tr}(\exp(-i t \Delta)) = \cdots \,,$

which is well known to yield terms that look very similar to various terms that appear in the action functionals for physical theories involving gravity and other forces.

Similar expansion formulas can be found for the cases where instead of a generalized Laplace operator we have a generalized Dirac operator sitting in a spectral triple. Instead of the above heat kernel we use

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

where $f$ is some regularizing function whose properties mostly drop out, $D$ is the Dirac operator and $\Lambda$ is some scale that we want to keep track of.

When $D$ is the ordinary Dirac operator on sections of a spinor bundle on some compact Riemannian space, the first order terms of the above expression reproduce the Einstein-Hilbert action functional describing general relativity.

This is in itself interesting, if maybe not shocking. What makes this approach really interesting, though, is that it admits a neat unification of the actions functionals for gravity and the other gauge forces.

Namely if we let $D$ be a Dirac operator as before, but now with respect to an associated spinor bundle on which we have an associated $U(N)$-connection $A$

(6)$D \mapsto D_A \,,$

then the above “heat kernel expansion” produces to lowest order not just the action principle of general relativity, but in fact that of general relativity coupled to the correct Yang-Mills action functional describing the gauge bosons given by $A$.

So this provides a neat way to encode all the forces encountered in the world entirely in the algebraic data provided by a spectral triple.

If this works for forces (bosons), it should also work for matter (fermions). And indeed it does - if we add one more term to our spectral action, one of the rough form

(7)$\langle \psi ,\; D_A \psi \rangle \,,$

for $\psi$ certain elements of $H$ (our generalized spinors).

In summary, the spectral action principle says that we should build action functionals $S$ for physical theories by picking spectral triples $(H,D,A)$ and writing

(8)$S(D,\psi) := \mathrm{Tr}(f(D/\Lambda)) + \frac{1}{2}\langle J \psi,\; D\psi \rangle \;.$

You can find details on this technique for instance in this review:

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

Once this idea was out in the world, an obvious quest was opened:

What is the spectral triple whose associated spectral action is that describing our world, i.e. that giving rise to the standard model action of particle physics coupled to the Einstein-Hilbert action of gravity?

It is not clear a priori what finding this spectral triple implies for our view of the world. If you are not impressed by games involving algebraic reformulations of otherwise well-understood concepts, you might not see more in it than a curious way to repackage information in a weird form.

On the other hand, it may happen that what looks weird afterwards is not the spectral triple, but the formerly so familiar standard formulation of the standard model it encodes…

More on that in a followup to this entry.

Posted at September 6, 2006 10:45 AM UTC

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Read the post Connes on Spectral Geometry of the Standard Model, III
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Read the post Connes on Spectral Geometry of the Standard Model, IV
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Excerpt: The details of the spectral triple which describes the standard model.
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### Re: Connes on Spectral Geometry of the Standard Model, I

hep-th/0610241
Date: Mon, 23 Oct 2006 11:16:35 GMT

Gravity and the standard model with neutrino mixing
Authors: Ali H. Chamseddine, Alain Connes, Matilde Marcolli

Posted by: Alejandro Rivero on October 24, 2006 3:02 PM | Permalink | Reply to this

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

Thisis, as promised, a discussion on the expected Chamseddine-Connes-Marcolli paper, hep-th/0610241, which is the consolidated version of the one we discussed before. It will is being presented in some detail during the NCG Semester in Newton Institute, and probably another talks around the word. You can get slides and audio of one of these talks, perhaps the central one, here.

The paper does a detailed travel from the newest formalism, based on an algebra $\mathbb{C}\oplus\mathbb{H}\oplus\mathbb{H}\oplus M_3(\mathbb{C})$ down to the pair of algebras $\mathbb{C}\oplus\mathbb{H}$ and $\mathbb{C}\oplus M_3(\mathbb{C})$ of the Red Book. This includes to consider (to introduce) the notion of Reality in the spectral triple (I suspect that it can be related to having quaternions around) and the notion of “unimodular” gauge potentials. Moreover, the explicit built of a curvature for the action, that in the Book forced a painful task of “junk removal”, is substituted with the concept of “Spectral Action” (plus Reality to do the full trick, perhaps). And in turn, having an Spectral Action allows us to put gravity in the same bag.

The Spectral Action is a regulated trace of the square of the Dirac operator (or of the Dirac operator with an even regulation function) plus a term for fermions, the later a variant of the old work on noncommutative geometry and reality, the former a new coming of the Chamseddine-Connes work. Fedele Lizzi has a conjecture that the term with the fermions comes from the trace action too but it is kept apart due to some irregularity in the density of eigenvalues of the Dirac Operator.

With respect to hep-th/0608226, the more striking addition is a remark (in section 2.7 and in the introduction, p. 4) about the “Moduli Space of Dirac Operators”, ie the possible values of Yukawa parameters. I can not avoid to think that this is related to a geometrical interpretation of the CP violating phases. This is because the more puzzling parameter just now is the number of generations, and CP violation is a way to ask them to be $\geq 3$, the fact of being 3 and not a greater number could then be argued from considerations on the functional integration. There are some remark about this in the audio of the talk, I believe.

Another puzzling remark appears in section 2.1 where complex and $3\times 3$ matrices are said to correspond to “integer spin” and quaternions to “half-integer spin”. This is in the spirit of considering that the algebra comes from a truncation at spin one of $\sum_n M_n(\mathbb{C})$, from $SU_q(2)$. Again, see the audio (???) of the talk. At some point Connes stops, chalks the summation, and asks the public “what is this”?.

Of the old axioms relating Spectral Triples to Non-Commutative Manifolds, Poincaré duality remains, but orientability is now under trial and perhaps discarded. This is a pitty, because a way to decide how to choose an algebra for the model was to check if the axioms for manifolds were working. On the other hand, it leaves room to try to discuss where this algebra comes from.

I find myself uneasy about the take on the predictions, supposed to be at GUT scale. I would very much prefer to aim for predictions at electroweak scale, because the more relevant physics for this geometry is the electroweak one. The work limps about this, on one hand the finite geometry supposed to be an approximation of some other object at high energy, on other hand the predictions been taken at GUT because, well, they fit when the renormalisation group runs them downwards. I’d like to point our that having the Weinberg angle of GUT is not by itself a strong indication of GUT, as its formula (the sum of hypercharges etc) can appear in other contexts. For instance, as the value of the Weinberg angle that minimizes $Z^0$ decay. On the other hand, while it is very encouraging that the paper gets a non susy formula relating the masses of bosons and fermions, I had enjoyed a lot more if the low energy, but 0.01 exact, relationship between Top mass and Higgs Vacuum had been derived from it.

Currently I am still trying to understand how asymptotical the asymptotic formula for the spectral action is. Ideally we should be able to switch off Newton Constant and then get the standard model, and to switch off the standard model couplings and get [a sort of] General Relativity. But this zone of the paper is a jungle of coefficients. Moreover, with thh standard model switched off, I would expect Gravity to come with the same proof that the theorem for commutative manifolds uses (see for instance the book of Varilly, figueroa and Gracia-Bondia). In fact the technique is the same, basically it amounts to ingenious use of the Lichnerowiz formula.

Posted by: Alejandro Rivero on October 26, 2006 8:46 AM | Permalink | Reply to this
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