The n-Category Café

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

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.

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.

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

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

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 $A$

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

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

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.