Connes on Spectral Geometry of the Standard Model, III
Posted by Urs Schreiber
I’ll try to outline the main technical ingredients that are involved in the computation of a spectral action # from a given spectral triple #.
Recall that we are imagining something along these lines:
We have a particle with worldline supersymmetry whose dynamics is defined in terms of a generalized Dirac operator
which is an odd-graded operator on a $\mathbb{Z}_2$-graded Hilbert space
on which an algebra
of observables is represented
by means of bounded operators on $H$.
More precisely, we want the $\mathbb{Z}_2$-grading on $H$ to be induced by an involutive operator
which is self-adjoint, squares to the identity, commutes with $A$ and anti-commutes with $D$.
Furthermore, we want there to be a real structure on $H$, which is an operator
squaring to $\pm 1$ and commuting and/or anticommuting with $D$ and $\gamma$.
In order to visualize this, you should
- think of $H$ as a Hilbert space of square integrable sections of a spinor bundle on a Riemannian spin manifold $X$;
- think of $X$ as the target space that our superparticle propagates on;
- think of $D$ as the Dirac operator of the superparticle – i.e. of $D^2$ as the operator measuring the energy of the superparticle;
- think of $A$ as the algebra of smooth functions on $X$ acting on $H$ by multiplication (this are our “position operators”)
- think of expressions $[D,a]$ with $a \in A$ as Clifford fields (e.g. $[D,a] = [\gamma^\mu \nabla_\mu,a ] = \gamma^\mu \nabla_\mu a$)
- think of $\gamma$ as what physicists call $\gamma^5$.
All taken together, we call this data a spectral triple,
and the point is that we think of any such triple of data satisfying some more or less obvious conditions as describing a superparticle propagating on a target space which is a generalization of the Riemannian target space $X$ used above.
Notice that
- $A$ encodes the topology of $X$;
- $D$ encodes the metric geometry of $X$.
In fact, Connes goes as far as identifying $D$ with the line element $ds$ on $X$ by writing
We associate two notions of dimension to such a generalized target space, called the
- metric dimension $d_\mathrm{met}$ determined by the decay rate of the eigenvalues of the operator $D$;
- and the KO-dimension $d_\mathrm{KO} \in \mathbb{Z}/8\mathbb{Z}$ determined by the choice of signs in the definition of the operator $J$ above.
We can think of the metric dimension as counting the number of “macroscopic” dimensions of target space of the kind we would measure by ordinary notions of movement in space.
And, while I am not completely sure at this moment, I believe we may think of the KO-dimension as the number of ordinary dimensions (modulo 8) of an ordinary Riemannian manifold which comes close to “approximating” our generalized Riemannian target space in some sense.
We shall be interested in finding spectral triples describing effective generalized target spaces $X$ that are good candidate models for the universe that we observe. This will make us want to have $d_\mathrm{met} = 4$ in order to get the basic properties of space and time right that we observe, and will make us want to have $d_\mathrm{KO} = 4+6 \in \mathrm{mod} 8$ in order to get the basic properties of the forces and particles we observe right.
But so far we have just a single superparticle propagating happily and undisturbed through an effective world $X$. In order for this to yield anything interesting as far as physics is concerned, we need to allow this particle to interact with itself.
Once it interacts with itself, we can regard our superparticle as a quantum of some quantum field $\Phi$ on $X$, such that that the dynamics of that quantum field are determined by the Feynman diagrams that describe the interaction process of our particle.
Instead of trying to write down these interactions explcitly, we shall at this point instead close our eyes, keep our fingers crossed and argue as follows:
“A sensible choice of interactions should yield a nice effective action $S$ for the quantum field. “Nice” means that it has a neat canonical expression in terms of just the data provided by our spectral triple. Moreover, we want the action to be additive under disjoint union of target space $X = X_1 \sqcup X_2$, which corresponds to $D = D_1 \oplus D_2$.
Taken together, this suggests that we set
(9)$S : (D,\psi) \mapsto \mathrm{Tr}(f(D)) + \langle \psi , D\psi \rangle \,,$where $\psi \in H$ is a (generalized) spinor field and $f$ is some function (and we use functional calculus to apply it to $D$).”
Keep your fingers crossed.
From the superparticle-with-interactions point of view that I have adopted here, the above is an attempt to guess the right form of the effective target space action without actually writing down the interactions. I feel motivated to adopt this point of view, because Ali Chamseddine has, long ago, made some necessary consistency checks which show that it has a chance of being justified.
Namely, if we think of our superparticle as being the point-particle limit of a superstring, then it’s interactions are fixed, and the effective target space action is known - and Chamseddine checked # #, to lowest order, that, indeed, it coincides with the expression $S$ above.
Be that as it may, we call $S$ the spectral action associated to the spectral triple. We want to Taylor-expand it and find spectral triples $(A,H,D)$ such that the first terms of this expansion reproduce the action functionals of Einstein-gravity coupled to the standard model of particle physics.
And we want to do so by the algebraic analog of a Kaluza-Klein compactification.
Given any effective target space encoded by a spectral triple as above, we say $X$ is a compactification on a $d_\mathrm{KO}$-dimensional internal space $F$ if
- the Dirac operator on $X$ is a tensor product $D_X = D_Y \otimes D_F$
- such that the spectral triple $(A_F,H_F, D_F)$ has vanishing metric dimension $d_{\mathrm{metric},F} = 0 \;;$
- and has the specified KO-dimension.
We address this as a “compactification”, because we imagine that there is an ordinary Riemannian manifold of dimension $d_\mathrm{ext} + d_\mathrm{int}$ such that its spectral triple is smoothly connected, in some sense, along a path through the moduli space of all spectral triples, somehow, with the generalized spectral triple described above.
So we imagine taking $d_\mathrm{int}$ of the original dimensions and making them gradually ever smaller and possibly ever more “non-commutative” in order to get from the higher-dimensional ordinary Riemannian space to the lower-dimensional exotic one.
Let $D_\mathrm{ext}\otimes D_\mathrm{int}$ describe a compactified effective target space.
We are interested in the case where $D_\mathrm{ext}$ is itself a combination of
- the ordinary Dirac operator $D_0$ of the ordinary Riemannian manifold that remains after the compactification;
- plus a part $\delta D$ that is a measure for the mixed terms of the metric of the internal and the external space .
Then, in total, the metric on our compactified effective target space will be determined by
- the standard metric on the external space encoded in $D_0$;
- the metric on the internal, compact space encoded in $D_\mathrm{int}$;
- the “inner fluctuations” of the metric, encoded in $\delta D$.
We will see in the end that
- $D_0$ determines the gravitational field on the effective compactified target space;
- $\delta D$ determines the gauge fields on that effective target space;
- and $D_\mathrm{int}$ encodes the metric on the internal compact space, which will manifest itself in terms of the coupling constants of the gauge fields
This is essentially precisely as in Kaluza-Klein theory for ordinary Riemannian spaces.
(Hm, and it looks as if I needed yet another entry to go into the details of the compactifications that we want to consider…)
Re: Connes on Spectral Geometry of the Standard Model, III
It would be great if you could go ahead to the point of describing the specific target space Connes uses for the Standard Model!
You know, it’s funny how everyone is adding 4+6 modulo 8 and getting 10 instead of 2.
The “mod 8” in KO theory is all about Bott periodicity. And, over the years, I’ve often emphasized how
10 = 2 + 8
hints that the target space and the worldsheet in string theory are related via Bott periodicity. The strongest evidence that it’s not just a numerological coincidence is how the 8 transverse degrees of freedom of a string in 10-space are neatly encoded using octonions. Properties of the octonions, or triality, then “explain” the supersymmetry. But, precisely this math is what gives rise to Bott periodicity!
To quote Robert Helling from “week104”:
I’ve also said many times that there should be a relation like “2d conformal field theories are to 3d topological quantum field theories as 10d string theory is to 11d M-theory”.
What’s interesting is that maybe some of the same numerology is showing up from a completely different viewpoint in Connes’ analysis of the Standard Model.
I made my own attempt to see the Standard Model as 10-dimensional; I have no idea if there’s a mathematical relation to Connes’ work.