The n-Category Café

If I may offer an opinion: You can use the German “Schnaps” for everything that is made of of some sort of grain and has more than 30% ethanol.

The Scandinavian may recognize this from the tradition of “drinking snaps”.

As for whisky: An aunt of mine once had a dog named Whisky (or Whiskey, I don’t remember), so, when I took him for a walk, I had to call “Whisky!” out loud on the street once, which prompted some curious reactions from other pedestrians - I demanded to know why they did not name him “Bobby”, but never got an answer…

I see, I misinterpreted Andrew’s comment. Too bad, I find watching Andrew in a language debate even more entertaining than having one myself.

As for Tim’s suggestion above, unfortunately the German word Schnaps is easily mistaken for the derivative English word schnapps, which doesn’t mean the same thing.

Urs wrote:

So gravity coupled to matter, as usually imagined, already is supposed to give a cobordism representation on unmarked cobordisms.

That’s what one might usually imagine, but it’s definitely not what people are studying in the papers I listed. Instead they are looking at pure 3d quantum gravity, without matter, and treating that as an extended TQFT. Then matter shows up automatically when one considers labellings.

(Technical subtlety: if we work with 3d Riemannian gravity with cosmological constant equal to zero, we get the Ponzano–Regge theory. Most of the papers I cited study this theory. This theory has some divergences that would need to be tamed to get a full-fledged extended TQFT. If we work with a nonzero cosmological constant with the right sign, we get the Turaev–Viro theory. This is an honest extended TQFT without divergences.)

What you are talking about is a curious step where we consider cobordisms with marked singularities on them that you want to think of as Feynman diagrams of matter fields. Curiously, on a single cobordisms with singularities, this would assign the fully quantized dynamics of the gravitational field, and at the same time, from what you say, encodes now a single Feynman diagram for the matter fields.

Right.

By the way, I only expect this to work in 3d quantum gravity, not 4d. In 3d, gravity has the special property that spacetime is flat away from matter. In 4d, gravity has propagating degrees of freedom, and things get a lot more complicated.

More later — gotta teach a calculus review session!

Okay, I’m back….

Urs wrote:

So to get from there to the actual quantum field theory of the matter fields, one would now have to do a sum of our representation of marked cobordisms over all possible ways to draw Feynman diagrams into them, renormalize that sum, Borel resum it.

Right.

What is the result of that?

Well, I don’t think anyone has actually carried out all these steps yet. I believe they have only obtained the Feynman rules — i.e. the rules for writing down a possibly divergent integral giving the amplitude of any fixed Feynman diagram. These rules depend on Newton’s gravitational constant $G_N$. At $G_N =0$ they’ve been shown to reduce to the usual rules for quantum field theory in Minkowski spacetime. For nonzero $G_N$ they give something new and interesting, which may describe 3d quantum gravity coupled to matter, assuming the further steps you describe can be carried out.

I urge you to look at this paper:

• Laurent Freidel, Etera R. Livine, Ponzano-Regge model revisited III: Feynman diagrams and Effective field theory.
  
  Abstract: We study the no gravity limit $G_{N}\to 0$ of the Ponzano-Regge amplitudes with massive particles and show that we recover in this limit Feynman graph amplitudes (with Hadamard propagator) expressed as an abelian spin foam model. We show how the $G_{N}$ expansion of the Ponzano-Regge amplitudes can be resummed. This leads to the conclusion that the dynamics of quantum particles coupled to quantum 3d gravity can be expressed in terms of an effective new non commutative field theory which respects the principles of doubly special relativity. We discuss the construction of Lorentzian spin foam models including Feynman propagators.

I am wondering: the Feynman-diagram-like singularities that you are looking at: are they really to be thought of as encoding matter, or not rather something like branes?

I can see why you’d want to call them branes, and that’s fine. Indeed, once upon a time I gave a talk at the Perimeter Institute on the 4d analogue of these ideas, which involves 1d extended objects — which I called ‘strings’ — coupled to 4d BF theory, instead of 0d extended objects (particles) coupled to 3d BF theory (= for example, 3d gravity). And afterwards, Malcolm Perry came up and said the word ‘1-branes’ would be better than ‘strings’ for these 1d extended objects. And he explained why.

However, the ‘0-branes coupled to 3d gravity’ really do act exactly like ordinary matter, at least in the limit where the gravitational constant $G_N$ goes to zero. So, people want to use them describe matter.

Something which might be related to this stuff is the following which has been confusing me lately: a 1-2-3 extended 3d TQFT is a 2-functor

where 3Cob_2 is the 2-category whose objects are closed oriented 1-manifolds, whose 1-morphisms are cobordisms between these, and whose 2-morphisms are diffeomorphism classes of cobordisms between those, and the 2-category 2Hilb can be replaced if necessary by your favourite “2-vector space”.

It’s pretty clear that the category $C := Z(S^1)$ assigned to the circle is a ribbon category. But suddenly this is somewhat of a paradox to me: we get out invariants of knots and links from a formalism which only apriori had to do with invariants of manifolds and cobordisms.

I know this sounds silly, after all this is what all that 3d TQFT / knots stuff was all about in the 1990’s… but if you look at the books (eg. Turaev, Bakalov and Kirillov) you’ll see that they don’t want to (or didn’t know how to) work with 2-categories of cobordisms, so they put in knots and links explicitly by hand into their formalism (they will talk about a “$C$“-extended 3d TQFT” by which they mean $C$ is some fixed semisimple category and “extended” means the cobordisms between the 2d surfaces come with knots and links sitting inside them). So it wasn’t surprising there: they put in knots and get out invariants of knots. But in our modern formalism of higher-categorical cobordisms it seems more interesting… we are getting out more than we explicitly put in (analogous to “putting in geometry gives you matter as well”).

I said:

So gravity coupled to matter, as usually imagined, already is supposed to give a cobordism representation on unmarked cobordisms.

John replied:

That’s what one might usually imagine, but it’s definitely not what people are studying in the papers I listed.

I understand that this is not what these articles are studying. What I do not understand is how what these articles are studying is “quantum gravity coupled to matter”.

To me it looks like what they are studying is a curious fact that some TFTs on cobordisms with embedded markings compute amplitudes of single Feynman graphs (and that for some QFT on Minkowski space, right!?).

Instead of giving me the feeling that suddenly this gives a deep explanation for how matter appears in pure functorial QFT, it confuses me: what on earth does it mean that a non-perturbative QFT that is allegedly a quantum theory of gravity computes single summands of some perturbation series of matter?

Not that I don’t find the fact interesting or remarkable, I just don’t follow how this fact can be read as giving “quantum gravity coupled to matter”. Can you tell my why you think that is the right way to describe this fact?

Urs wrote:

To me it looks like what they are studying is a curious fact that some TFTs on cobordisms with embedded markings compute amplitudes of single Feynman graphs (and that for some QFT on Minkowski space, right!?).

No, not for a quantum field theory on Minkowski space! For a quantum field theory of a novel sort, where the motion of the particles affects the geometry of spacetime.

Only in the the limit where the gravitational constant goes to zero do the Feynman amplitudes reduce to the usual thing you’d get for a quantum field theory on Minkowski space. That’s just a consistency check, though a remarkable result in itself.

The interesting part is that when the gravitational constant is nonzero, the particles affect the geometry of spacetime exactly as you would hope for in a theory of 3d gravity. You can compute gravitational holonomies around the particle’s worldlines and they work out exactly right.

You can compute gravitational holonomies around the particle’s worldlines and they work out exactly right.

Sorry for being a pain on this point, but I still don’t understand why in quantum-$\langle$gravity coupled to matter$\rangle$ it should make sense to speak of any worldlines of particles at all.

If there is a fixed worldline of a single particle, then the matter field content of which that particle might be a quantum has not yet been quantized.

And my confusion goes further: even if I assume that I did understand what it would mean to have a classical particle trajectory in a quantum gravitational background, then I would be hard-pressed to see what gravitational holonomies around it were.

I would know what gravitational holonomies of a classical gravitational field background around a line-like singularity would be. But somehow the claim is that we are speaking about quantum gravity coupled to quantum/classical(?) matter here, instead. Which confuses me.

Urs wrote:

Sorry for being a pain on this point, but I still don’t understand why in quantum-⟨gravity coupled to matter⟩ it should make sense to speak of any worldlines of particles at all.

Of course you should already be worried about this in quantum field theory. And indeed, Bohr didn’t like it when Feynman first drew those famous diagrams, because the edges of those diagrams seem to describe particles having definite positions at each time, and that’s not how it works in quantum theory. But really, of course, these diagrams are symbols standing for the process of integrating over all possible worldlines. And that’s roughly what’s going on here, too. But now of course it’s even more tricky to interpret.

So perhaps I shouldn’t have said ‘worldline’: perhaps I should have said ‘Feynman diagram edge’. In fact it’s probably dangerous to use any pre-existing terminology for a theory of this new sort. So let me try to say some things of a more mathematical character:

For any value of the gravitational constant one can calculate an amplitude for a Feynman diagram embedded in a 3-manifold, with edges labelled by masses and spins and with vertices labelled by intertwining operators. You can interpret this amplitude as an integral over geometries that are flat away from the Feynman diagram, and have specified conjugacy classes of holonomies around the diagram’s edges — where these conjugacy classes are specified by the masses and spins. But in fact these amplitudes can be computed in a purely algebraic way starting from $Rep(SU(2))$.

In the limit where the gravitational constant goes to zero, these Feynman amplitudes reduces to the usual Feynman amplitudes for a quantum field theories on 3d Euclidean space. Otherwise you get corrections depending on the gravitational constant.

You can also add extra observables corresponding to Wilson loops around the Feynman diagram edges. The expectation values are what you’d expect from a connection that was flat away from the Feynman diagram edges and singular along the edges, with the singularity you’d get from a classical particle with the specified mass and spin.

I should perhaps again emphasize that we only expect something so simple to happen in 3d quantum gravity, where spacetime is flat except where there’s matter. This is why, for example, the mass and spin of a classical particle determines the conjugacy class of the holonomy around its worldline! It’s also why we can smoothly deform the worldline of the classical particle without changing the holonomy around a loop that encircles that worldline. Neither of these things would be true in 4 dimensions… and these things make the path integral computation of the Wilson loop expectation values infinitely simpler than they would be otherwise. So if your intuition is based on the more interesting higher-dimensional theories, everything I’m saying should seem completely crazy.

Ok, thanks for this. This is an aspect of the story I haven’t appreciated properly, though it has indeed been stressed eg. in Jeffrey’s thesis.

Ah, this is the sort of discussion I enjoy at this place … I’m sitting in the sitting room of a beautiful german Schloss, and am a little bit too schlosshed myself to make any useful contribution to this. But I do remember when Jeffrey explained this stuff to me a few years ago when we met in Nottingham once, and again when we met at the PI — it’s beautiful!

This stuff gives you a way to turn string diagrams into 2d cobordisms, and so gives a nice way to turn ‘internal’ things in your modular tensor category into ‘external’ things.

Don’t get so schlosshed that you fall out of the Schloss!

By the way, regarding that slide from Jeffrey’s talk… irreducible representations of $SU(2)$ are labelled by spins, so you might wonder why he’s talking about ‘mass and spin’.

The point is that in the Ponzano-Regge theory, the category for the circle is not $Rep(SU(2))$ but the center $Z(Rep(SU(2))$, which consists of representations of the ‘quantum double’ of $SU(2)$. Irreducible representations of this gadget are labelled by masses and spins.

Again, it’s a special feature of 3 dimensions that the (double cover of the) Euclidean group is

$$SU(2) \ltimes \mathbf{su}(2)$$

and thus has representations that are rather similar to those of the quantum double of $SU(2)$.

Indeed, Freidel and company show how the quantum double can be thought of as a ‘$G_N$-deformed’ version of the Euclidean group, where $G_N$ is Newton’s gravitational constant. As I explained in week232, this deformation has the effect of putting an upper limit on the mass of particles — roughly the Planck mass!

In addition to this $G_N$-deformation we get a further $q$-deformation when we turn on the cosmological constant.