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 – $\mathbb{R}^n$ 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 $U_q({sl}(2))$ (a q-deformed version of the universal enveloping algebra $U({sl}(2))$, which for $q=0$ is the algebra where the Lie bracket of ${sl}(2)$ 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 $G$ – (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 “$\dagger$-monoidal category”. Not incidentally to all this, he also shows that in finite dimensions, at least, a $\mathbb{C}^{\star}$-algebra is “the same thing as” a $\dagger$-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.