The n-Category Café

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”

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

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

(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.