Well, a week ago I got back from England, where I spent a week at the University of Nottingham at the conference “Quantum Gravity and Quantum Geometry 2008″, and a weekend visiting friends in London. London was enjoyable, though surprisingly expensive. It’s strange, when so many things are traded globally, that prices differ so much from place to place – the standard rule being to imagine that all prices in Pounds are actually in dollars, and they seem quite familiar. Clearly not everything is affected by trade, with restaurant meals among them. In any case, it was quite interesting to come come from London, Ontario to London, England, and walk around all the places whose names show up attached to completely dissimilar landmarks in the Canadian version.

As for the conference, it was a great experience. This was an outgrowth of the “LOOPS” series of conferences. The only one of those I’d been to previously was LOOPS ’05 at the Albert Einstein Institute, in Germany. At that time the conference was a little more focused on some particular approaches to quantum gravity (though there was still a whole range of talks). This year, there seemed to have been some attempt to broaden the conference a little – one result being that there must have been about 200 people attending, with something on the order of 90 talks, most of them half-hour talks in the parallel sessions. As a result, I saw less than half of what was going on. However, there were some broad subject areas, such as loop quantum gravity, spin foam and combinatorial quantization, noncommutative geometry, quantum groups, as well as some less readily classifiable talks.

In one talk on the first day, Carlo Rovelli discussed the relation between the Loop Quantum Gravity and spin-foam approaches to a theory of 4D quantum gravity. In particular, he was talking about the fact that the two approaches agree with each other in 3D, but it’s not so clear they do in 4D – or at least, it’s not clear what the spin foam model is that does this in 4D. This is part of what’s behind the program to improve the Barrett-Crane spin foam model for 4D gravity. It has various technical problems as well, which various more technical talks got into in more detail later in the conference. Rovelli was describing work on the new models which agree with LQG. Various other people have done work on this, including (among others) Freidel (who talked about that in his own talk later) and Krasnov, and Engle, Pereira and Rovelli. Florian Conrady also talked about these new models later on. I know Igor Khavkine, just graduating here at Western, has also done some work on these.

Another talk based off the successes of these models was by Abhay Ashtekar, about Loop Quantum Cosmology – that is, applying loop QG methods to the universe as a whole – a quantum version of the Friedman-Robertson-Walker universe. What’s interesting about this is that they’re doing numerical and analytic simulations, and predicting something that otherwise has usually been added as a “what-if” afterthougoht. Namely, such a universe behaves a lot like classical FRW, except near the “big bang”, classically a singularity, where quantum geometric effects prevent that from happening. Continuing through the other side, one sees a collapsing universe – an overall “bounce” effect. An interesting prediction, if hard to check.

In any case, I was bombarded by a whole range of other talks on other points of view. Starting from the very first talk, by Vincent Rivasseau, there were several talks presenting noncommutative geometry, Alain Connes-style, as a setting for a quantum theory of gravity. There’s certainly an appeal to the idea of replacing measure-theoretic and topological information about spacetime with a quantum algebra of observables – just write the theory in quantum terms from the start, giving up the usual differential geometry for its noncommutative version. Rivasseau presented, among other things, the idea of QFT as weighted species, in the sense of Joyal’s combinatorial species. I thought this was great, since I looked at just that idea for the simplest QFT of all, the quantum harmonic oscillator.

(Speaking of which, I had some interesting conversations with Jamie Vicary in which I finally “got” part of what he did with his own paper about the oscillator – which is to show how “taking Fock space” for a quantum system is a monad, namely the monad associated with the “free commutative monoid” functor, and its adjoint.)

Shahn Majid, whom I knew as the author of some well-known books on quantum groups, also spoke about this C*-algebra approach to geometry, and quantum gravity. : begin with a space, like a manifold, or better yet a fibre bundle, which is where a lot of physics gets done, and look at the algebra of forms on it. It has nice properties (it’s a differential graded algebra, etc.), including being commutative. One can deform these to noncommutative algebras that are quite nice – “q-deformation” assumes the commutators between elements depend on some parameter q, so the old picture where q=0 is simply a special case.

So then one thing is to develop a deformed version of classical things from geometry and analysis – for example, the Fourier transform. Even in the big purple book on quantum groups, he outlined what this approach consists of: a criterion for a quantum theory of gravity, that it should be algebraically “self-dual”, under exchange of “position” and “momentum” variables. (That is, under a Fourier transform – being its own Fourier dual).

Well, speaking of quantum groups, I should mention Aaron Lauda’s talk on categorifying them – specifically, on categorifying “deformed classical Lie groups”, like (a q-deformed version of the universal enveloping algebra , which for is the algebra where the Lie bracket of is a genuine commutator). He described a graphical calculus – a particular kind of string diagram, with some relations on them – which is a categorification of the quantum group. In fact, as sometimes happens, it categorifies a specific presentation of the algebra in terms of some generators and relations.

An appealing thing about these string diagram methods and so forth is that it suggests why these algebraic gadgets – quantum groups, in this case – are good at encoding topological information about tangles, braids, knots, and so on. If diagrams that involve those shapes categorify (read “model the underlying structure of”) quantum groups, then it makes sense that quantum groups to give invariants for them.

Along similar lines, Joao Faria Martins talked about invariants for “welded virtual knots”, and for knotted surfaces from crossed modules (read “2-groups”, if you’re so inclined – they are equivalent). Martins also published a paper with Tim Porter about related work, which in turn builds on David Yetter’s, on a class of manifold invariants. Their paper talks about “extending the Dijkgraaf-Witten model to categorical groups” (Urs Schreiber, possibly among others, rephrased that to call it a “categorification of the Dijkgraaf-Witten model”. The DW model is the TQFT foundation for my own look at extending (read, “categorifying”) TQFT’s based on gauge theory using a group – (finite, for the DW model). These are categorifications in two different directions, though: one, from a gauge group to a gauge 2-group, the other from a TQFT – a functor – to a 2-functor given by a group. Probably for 4 dimensions and higher, the 2-group version or higher is the most interesting to study.

In fact, there was a fair bevy of talks relating to categorical methods in quantum geometry. For example, Jamie Vicary gave a talk introducing a “categorical framework for quantum algebra”, by means of non-threatening string diagrams. These can be used to show the axioms for a “-monoidal category”. Not incidentally to all this, he also shows that in finite dimensions, at least, a -algebra is “the same thing as” a -Frobenius algebra.

Benjamin Bahr gave another talk dealing with categorical issues – namely, how to get measures on certain groupoids, such as, indeed, the groupoid of connections on a manifold. In fact, he treated various cases under the same framework: flat and non-flat connections, on manifolds and on graphs – and others.

In all, I was pleasantly surprised by the mix of the physically and mathematically inclined points of view, and the trip itself was a lot of fun.

July 17, 2008 at 8:38 am

Hi,

thanks for the report!

I am thrilled by this bit:

I had not heard of Bahr’s work before. A quick search on the web didn’t yield anything (but maybe I did’n search carefully enough). But I am very interested in this general question. Have thought a bit about it myself.

Do you happen to have any further details on this? Is there anything written available in any form?

Do you know if his groupoid measures are at all related to Tom Leinster’s weightings on categories?

Does Bahr go as far as saying something about path integrals for gauge theory? Does he mention connections to BRST-BV methods which are the standard tool for handling integration over things like the groupoid of gauge connections?

For the case of finite groups and hence necessarily flat connections, the groupoids they live in are, naturally, well understood. More strikingly, it is known in these cases that the right “path integral” (just a sum in this case) measure on these groupoids — which appears in Dijkgraaf-Witten theory but also in its higher version, the Martins-Porter model that you mention above — is precisely the Baez-Dolan-Leinster groupoid measure for DW, and its higher version for the Cat-enriched case for the Martins-Porter model. (I recall this for DW and prove it for 2DW in section 1.4 here).

Does Bahr’s approach says anything about this? Does he reproduce the measures for DW and higher DW theory? Do you happen to know?

I’d be grateful for whatever information you might have.

July 17, 2008 at 4:43 pm

Hi, Urs:

I was also very happy with Benjamin Bahr’s talk. I asked him if there was an associated paper where I could look at the details, since his talk was after all only 25 minutes and fairly introductory. He said that there was one coming shortly, but the full paper will be out later this year, IIRC.

However, he does have some other work online, most recently a bunch of papers with Thomas Thiemann, including this one, which is not what his talk was about, but seems to touch on a few related issues, in that he was talking about functors from path groupoids.

What I can recall about his the is that it dealt with methods that for building a topology and measure on functor categories like and – or indeed one could start fairly general groupoids, like the path groupoid of a graph. (Note that he’s not just talking about smooth functors, here, where that’s relevant.)

The idea is that these groupoids – locally finite ones – can be represented as the projective limit of a poset of certain finitely generated, groupoids. The functor category you want is then also a projective limit, and gets a topology and measure on it from that.

I have the impression the paper he mentioned contains more good stuff, but of course it’s not out yet.

July 18, 2008 at 5:17 am

Is there a link to a paper on this idea of Vincent Rivasseau? Will slides be put online?

July 18, 2008 at 7:59 am

I believe slides of (most of) the talks will be put online – the organizers were certainly asking for copies to be uploaded by a certain date – but so far I haven’t seen them on the website.

As for Vincent Rivasseau’s talk, the title was “Noncommutative Renormalization”, and significant portions of the talk correspond to the material in this paper. The discussion of QFT in terms of weighted species does not, but the way he talked about that led me to think it was some background material that was not original to him, though if he gave a reference for it, I didn’t write it down. I remarked on it because (a) I like that idea, and (b) I haven’t seen others discuss it.

What I did write down of that part of the talk says roughly the following: A QFT can be seen in terms of generating functions of weighted species (with some parameters such as coupling constants). He cited Bergeron, LaBelle and Leroux’s book “Combinatorial Species”. The weights in these species may be given by divergent series, which is a job for renormalization. (Constructive field theory, on the other hand, deals with the possibilty that the generating function may itself be a divergent series – here he made a remark which I didn’t understand, but wrote as “Species of Graphs Species of Trees”).

The remainder of the talk described some combinatorial rules for doing renormalization of a theory whose interactions are given by Feynman graphs. The operations involved shrinking parts of the graphs into new vertices, with new amplitudes. I don’t know a lot about renormalization, but this stuff looked familiar.

Then he gave a noncommutative example, for a theory on the Moyal space , which was similar in flavour, and I take to be the new part.

July 18, 2008 at 11:26 am

Hi, this is really very inspiring web site ! …

I find very interesting the entry Correspondences and Spans … since I kind of work in similar things … hope to have more time to return on this later ;-)

I was at the QG2-2008 too … and it is a great pity that we just did not find a chance to talk a bit together there …

As regards the slides (and some of the videos) of the talks at the QG2-2008 conference in Nottingham, they are all on-line now, but well hidden here.

;-)

Best Regards. Paolo

July 18, 2008 at 11:59 am

Hi,

I had taken the liberty of forwarding some of this to the n-Café. There Paolo Bertozzini pointed out that slides for most of the talks are available here.

In fact, the slides of Bahr’s talk are here. Apparently this is not quite what I was hoping for (I have comments on them here).

But then, John kindly dropped a brief comment indicating that Alan Weinstein has figured out exactly what I was hoping for. So now I am looking forward to seeing Weinstein’s idea revealed to the public. :-)

July 29, 2008 at 5:17 am

Having looked a little further, although the work of Rivasseau is extremely interesting, it appears that he refers to

speciesin a general combinatoric sense. I see no evidence that he is familiar with concepts of Joyal et al.July 29, 2008 at 5:21 am

Hmmm. No, wait. I didn’t realise that Joyal was responsible for the whole subject, going back decades, but now wikipedia suggests that this is the case. Cool.