Manuel Bärenz and I have just finished a paper on 4-dimensional topological state sum models. It is posted here (since 12 Jan 2016), since arXiv can’t cope with typesetting the diagrams yet. The idea of the paper is to squeeze more out of the Crane-Yetter state sum model and perhaps indicate how to get a viable quantum gravity model (or solid state physics model) from it.
The original CY was defined just using the quantum group version of SU(2), and turns out to be rather too simple be an interesting physics model. The reason is that the group SU(2) is used to “colour” both one- and two-dimensional dual edges. This has the effect that the quantum theory can’t “see” the difference between the one- and two-dimensional stuff.
All this is best seen using a handle decomposition rather than a triangulation. There’s an operation that changes 1-handles into 2-handles (thus changing the topology of the manifold) and the problem with the original CY is that it is invariant under this operation – which is a property that isn’t wanted. As a consequence, CY is the same on lots of different manfolds, which is why it is “too simple”.
One of the things that Manuel and I have is an efficient translation between the triangulation picture and the handle picture. In the handle picture, Jerome Petit had the idea that the 1- and 2-handles can be coloured differently, to give a new set of models that he called “dichromatic”. We have understood that in the triangulation picture, this corresponds to the CY invariant being “nonmodular” (the original CY is “modular”). Interestingly, there are also models in the handle picture that don’t have a CY description at all. We calculated a few simple examples and found that one of them has configurations that are a plausible analogue of “teleparallel gravity” in the formulation given by Baez and Wise. This doesn’t yet mean we have a new quantum gravity model because, firstly, we only used finite groups instead of Lie groups (to keep things simple) and secondly, it isn’t clear that the action will be the gravity action. Still, it is an interesting direction.
What it needs next is to do a lot more examples. Probably any really interesting examples will involve representations of a group or quantum group that are non-unitary. New territory indeed!
Lisa Glaser and I have started to investigate random non-commutative geometries in our latest paper
Monte Carlo simulations of random non-commutative geometries. The geometries are all specified by a Dirac operator that is a finite-dimensional matrix, so in this theory the partition function is a finite-dimensional integral. We investigated the phenomena that occur by calculating this integral on a computer, in much the same way as you would measure data from an experiment in a lab.
The most fascinating finding is that by tuning a parameter in the action we are able to find a phase transition (audible here!), and at this transition the spectrum of the Dirac operator looks a lot like the spectrum one would expect for a typical manifold in the usual sense of differential geometry. This is surprising, because non-commutative geometries generally look nothing like differential geometries.
The first plot shows the eigenvalue spectrum for the fuzzy sphere using an algebra of 10×10 matrices. The plot has the eigenvalues of the Dirac operator on the horizontal axis and the density of eigenvalues vertically.
The spectrum for the fuzzy sphere is the same as for an ordinary two-sphere in differential geometry, except that the spectrum is cut off after a finite number of eigenvalues (in this case, 200). These spheres are special in that the eigenvalues are in fact all integers, and have a multiplicity proportional to the eigenvalue, due to the spherical symmetry.
A Dirac operator on an arbitrary metric on a surface would not have this symmetry or the degeneracy, but the approximate V-shape of the plot should be the same, at least when the eigenvalues are large enough.
This is exactly what one sees in the second plot, an approximate V-shape with some slightly different detail for small eigenvalues. But the second plot is generated by an ensemble of random non-commutative Dirac operators!
Actually, we nearly didn’t find this. We looked at a range of different random NCG models with a free parameter and accumulated a large number of eigenvalue plots that looked nothing like a manifold. Only when we started to write a section of the paper explaining this did it become clear that a very small number of the plots did look something like a manifold. These plots occur only close to a phase transition, where the properties of the ensemble change sharply when the parameter in the model is changed.
Our findings fit into a general theme for discrete theories, like lattice gauge theory, where continuum-like behaviour occurs at, or near, a phase transition. Thus it seems like a great place to look for quantum gravity in non-commutative models.
The idea behind the project outlined in my Como lectures is to model a space-time geometry with a (finite-dimensional) matrix. Then quantum geometry (or random geometry) can be modelled by a suitable integral over the space of matrices, which is again a finite-dimensional integral, and thus well-defined.
In general terms, integrating over a space of matrices has been studied a lot, and goes by the name of a “matrix model”. However, in this case the matrix variable is a very specific type of matrix, namely a Dirac operator. The Dirac operator satisfies a number of algebraic conditions, so my new quantum gravity can be thought of as a matrix model for a very specific type of matrices.
The point of using Dirac operators is that a Dirac operator specifies a geometry. In the classical case, the Dirac operator is a differential operator on a manifold and one can extract the metric tensor from it. In the case of a matrix, the number of spinor “modes” that the Dirac operator operates on is finite, so that there is a maximum energy. Physically, if there are matrix quantum geometry models that behave like quantum gravity, this maximum energy should be identified as the Planck scale. So the mysterious Planck-scale cutoff that is needed to make sense of quantum field theory is built in to the model in a fundamental way. Whether there really are viable matrix quantum gravity models is very much an open question, and the point of The Project is to try to answer this.
To define a matrix Dirac operator in a consistent way, the notion of geometry is generalised to “non-commutative geometry”. In this setting, a lot of geometric constructions still make sense. One hopes, for example, that there is a suitable notion of the limiting case of large matrices in which a sequence of matrix Dirac operators converges to one of the usual Dirac differential operators (“taking the Planck scale to infinity”). More loosely, one can understand constructions in non-commutative geometry as the algebraic analogues of familiar constructions in ordinary metric geometry on a manifold. So it is important to have all of the machinery of non-commutative geometry (technically, the Dirac operator is part of a “real spectral triple”).
The basic set-up for matrix geometries is explained in my paper “matrix geometries and fuzzy spaces as spectral triples”. The paper contains the general definitions and a lot of useful data, such as operations on gamma matrices and explicit forms for the Dirac operator. I’m hoping it will be a resource for future efforts at modelling quantum geometry.
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.
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.
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.
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).