## Como lectures

I’m giving some lectures at the SIGRAV school in Como on “Non-commutative geometry and quantum gravity”. I’ll put the slides up here as they happen.

Lecture 1: The Planck scale. This lecture reviews the experimental evidence for a fundamental Planck scale. Various theoretical approaches to modelling the Planck scale are mentioned very briefly, mainly concentrating on the predictions of the spectral action suggested by non-commutative geometry.

Lecture 2: Non-commutative geometry: axioms, examples. The axioms of non-commutative geometry are given in this lecture alongside an explanation of how they are satisfied by an ordinary (commutative) manifold. The example given by the standard model internal space is also summarised.

Lecture 3: Matrix geometries and fuzzy spheres. The lecture explains my project to replace the usual space-time manifold with a non-commutative geometry based on finite-dimensional matrices. Examples with spherical symmetry are considered.

Lecture 4: State sum models, using 2d examples. A different topic is introduced in this lecture: 2d topological state sum models defined using a diagrammatic calculus. Some recent progress in defining models that are sensitive to the spin structure of the surface (and are thus fermionic) is explained.

Lecture 5: Non-commutative geometry in 2d state sum models. The final lecture connects the topic of state sum models and non-commutative geometry. It shows how defects in a 2d state sum model can carry a non-commutative geometry.

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Giving the lectures turned out to be enormously useful. I’m writing a paper called “matrix geometries and fuzzy spaces as spectral triples“, which ~~I hope will appear soon~~ *is out now*. The paper will be much better for having had to explain the ideas in the lectures.

## Hope and glory

In a given field of research, there’s a big distinction to be made between the scientific achievements in that field and the future prospects for the field. The “glory” is a matter of record, the stuff of review articles, whilst the “hope” is a matter of opinion. So what’s the role for opinion in scientific research?

In very pure mathematics there’s a presumption against expressing your opinions (in print anyway) in case it misleads readers, who would rather construct their own (and perhaps very different) picture of how the subject might develop. This certainly means that opinion and fact are never confused, but the problem is that there is consequently no debate about the aims of a subject or about the value of what has been achieved.

One might think that opinion has no place in science but that isn’t really right. Treating every possible avenue of future research as equally worthy of attention is not what we do; it would be a good way to waste resources. Every scientist has to make a judgement about what is worth doing and what is not. So we have opinions on the way forward, and also often share them when we communicate science. It seems to succeed in a subject it is necessary to be optimistic about its prospects, and to convince others that what you are doing is worthwhile you have to communicate that optimism.

My view is that expressing that optimism is quite legitimate but one should be careful not to let it be confused with the story about what has been achieved.

So where are we in quantum gravity research? As far as the “glory” is concerned: nowhere. Approaches based on quantum field theory have lots of “approximation” schemes but no central definition (so one cannot say they are actually approximating anything). Superstring theory does not even have a complete perturbation series, and a general non-perturbative formulation does not exist. Loop quantum gravity does not exist either; while there is a description of a state space that is “too big”, there is no definition of the constraints that should reduce this space to the physical states. Discrete approaches such as spin foam models avoid troubles with the continuum, but the models that have been investigated so far are not supposed to be a complete physical theory; there have both technical and conceptual shortcomings.

As for the “hope”, there are plenty of reasons to continue to work in these fields. For example, superstring perturbation theory is the subject of current research and there are grounds to hope that a more complete perturbation theory will emerge. And maybe this will help in looking for the non-perturbative theory. String theorists could do themselves a favour by dropping the pretence that a complete string theory exists. After all, how can you recruit people to work on this problem if the wider scientific community thinks it has all been done already?

However I’m not going to get excited about superstrings because experimental tests of supersymmetry have a habit of coming out negative. Whilst this does not *prove* the absence of supersymmetry, in my opinion it is not looking very likely.

I’m continuing to work on the overlap between higher category theory and quantum physics. Some of this is work on understanding the spin foam models of quantum gravity. But I think there is more physics to be extracted from higher category theory. Indeed the mathematics is already deeply involved with both topological quantum field theory and conformal field theory, both of which are mathematically well-defined.

Moreover, there is a lot of unexplored territory. For example, there are “defects” in topological models that are point-like, string-like, brane-like etc., and there is a role there for deformations of geometry that are non-commutative, non-associative or other such things that don’t even have names yet. There are non-commutative geometries in particle physics that look a whole lot more enticing than supersymmetry. How do these examples fit into categorical quantum field theories? I hope to find out.

## Gray category diagrams

The usual spin networks are 1-dimensional networks (vertices and edges) embedded in a 2-dimensional space, and decorated with objects and morphisms in a tensor category.

What does this look like in one higher dimension? The networks are now 2-dimensional complexes, having vertices, edges and surfaces glued together. These are embedded in 3-dimensional space and labelled with data from a suitable type of higher category called a Gray category (after John Gray). Each 3-dimensional region in the diagram is labelled with an object in the category, each two-dimensional surface with a 1-morphism, lines with 2-morphisms and vertices with 3-morphisms.

In a recent paper, Catherine Meusburger, Gregor Schaumann and I worked out a theory of these *Gray category diagrams*. When I’m around physicists, I’d like to call these *spin 2-networks*, but I probably won’t yet, until a few things have been better understood. So I’ll explain what we did find, and then what else could perhaps be done.

The first thing to mention is that to get the most interesting theory, the Gray categories should have duals. Without duals, the diagrams are directional, but the duals allow lines and surfaces to turn around. This is familiar in spin networks, where the duals supply the “maxima” and “minima” for lines. The axioms for duals are (more or less) those given by Baez and Langford. However we discovered an extra condition is needed to get invariance of the diagrams under mappings of three-dimensional space. We called this condition the *spatial condition*, and it is in fact a generalisation of Kauffman’s double twist move for ribbon graphs (a ribbon category is an example of a Gray category). So a *spatial Gray category* has diagrams that are invariant under mappings of 3-dimensional space.

We did lots of work to understand the spatial condition: it comes about in comparing functors of Gray categories that implement the duality operations that are, roughly speaking, rotations through pi around one of the axes of the diagram. Also there are some reasonable algebraic conditions that guarantee that a given Gray category with duals actually is spatial.

There are a couple of things missing in our presentation. The first is a technical lemma that goes into the proof of the invariance. In knot theory, the first two Reidemeister moves plus Kauffman’s double twist move are sufficient to prove spatial invariance. In our setting, we need a generalisation of the Reidemeister moves to to allow moves on the surfaces as well. These moves are well-known for smooth surfaces but we are working with PL surfaces, and, frustratingly, it seems that the corresponding problem has not been studied. It is hard to believe that the solution will be anything but the same, but someone should sort this out.

The second thing is a good theory of framings for the diagrams. In knot theory, a framing is a choice of a “frame” at each point on the knot, which can be thought of as a coordinate system for the tangent space. Such a frame is determined by thickening the knot to an oriented ribbon; the frame is given by the direction along the knot, the normal to the ribbon and a third direction orthogonal to the first two. In principle it seems possible to define a framing for Gray category diagrams by thickening up the diagrams in an analogous manner. I tried this but in practice it turned out to be horrendously complicated and it did not make the paper. It seems that some things that work in low dimensions aren’t necessarily the right thing to generalise to higher dimensions. It is possible to work without framings (essentially, everything is “blackboard framed”), but it would be more geometrically elegant to have some workable notion of framing.

Finally, what is this all good for? The original motivation was to understand defects in 3(=2+1)d topological state sum models, which this goes a long way towards. The other “physics” motivation is to provide the data for constructing 4d state sum models (building on the work of Mackaay).

## A fermionic state sum model

Recently Jorma Louko, Steven Kerr and I produced a state sum model with fermionic variables. It is only in 1d space-time, so something of a toy model, but it illustrates some important principles. Firstly, the model is triangulation-independent, which means it has the “diffeomorphism symmetry” of general relativity. Secondly, the fermion integration induces an action for the bosonic gauge field (as expected) but without a cosmological constant (very unexpected). I’m still somewhat puzzled over exactly how that works, but it would be very nice if there are models like that in higher dimensions.

## The Spin Foam Lectures (UNAM)

The lecture slides are available on the page Talks: slides and videos.

## David Barrett

My father died last summer – hence the big gap in posts while I catch up with everything else. There’s lots of things I could say about him – his love of games and puzzles for example. But here I just want to explain one thing: he taught me my first mathematics at the age of nine or ten, some trigonometry and algebra. I can remember finding mathematics at school easy and I am sure that what he taught me helped greatly. It wasn’t so much knowing the trigonometry and algebra that was so helpful, more that I understood from him *how to go about doing mathematics.* That seems to me to be a very important part of teaching.

## MSc course

Here at Nottingham we are running an MSc in *Gravity, Particles and Fields*. This is very specifically aimed at students interested in getting into relativity and particle physics research, in areas such as quantum gravity, cosmology, quantum information, etc.

The MSc runs from each September and applications can be made online at any time. More info

**Update ** There are some grants for well-qualified students, the details depending which country you are from (or currently in). Details are under the funding tab.