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).