So last time I was describing this “matter without matter” idea and claiming that it has something to do with TQFT and the Ponzano-Regge model of quantum gravity. I’d like to get a little more detailed here.

To describe this in physics terms, it’s easiest to understand the point if, instead of using the (more technically accurate) terms “manifold”, “cobordism between manfolds”, and “cobordism with corners between cobordisms, I name-drop the terms “boundary”, “space”, and “spacetime”. But the caveat here is that these terms really imply a certain geometric structure which I’m not actually assuming is there: a specific geometric structure on these manifolds is a state of the theory. Furthermore, with Ponzano-Regge, we’re talking about Riemannian gravity – there’s no such thing as a “timelike” direction. So using the term “spacetime” is being rather optimistic that everything will work out in more physical settings – but it’s a helpful motivation.

At any rate, the way I describe it in the thesis, in $n$ dimensions the typical setup for an extended TQFT in the sense of a weak 2-functor into 2-Vect, one has “boundaries”, which are manifolds of $n-2$ dimension (in 3D, each boundary is some union of a bunch of circles, and in 4D it would be a union of surfaces, each with some genus). These are joined by “spaces” (cobordisms), of $n-1$ dimensions, which are in turn connected by “spacetimes” (with the above caveat). These cobordisms are, in particular, cospans in some category of spaces, and they give rise to spans of groupoids of configurations for a gauge theory.

In any case, how does this relate to gravity? The answer is by way of topological gauge theory: the extended TQFT in question has a lot to do with flat connections on manifolds $M$ (or indeed manifolds with boundary or corners), which is what topological gauge theory is about. One way to say what a flat connection is, is to say that it takes a path in the space $M$, and gives an element of the gauge group $G$ (this is not the most well-known way to describe a flat connection – more on that in another post, but I’ll cite weeks 8 and 9 of the spring 2005 UCR Quantum Gravity Seminar for now).

If the gauge group $G$ represents the symmetries of something we’re transporting around the surface, this tells us how that thing is being transformed as we move it. For gravity, we take the gauge group to be the symmetries of a model spacetime – what spacetime “looks like locally”. For standard special relativity, this is the Lorentz group $SO(3,1)$ – the symmetries of Minkowski space. For 3D gravity, it’s $SO(2,1)$ (symmetries of Minkowski space with two space and one time dimension). For 3D Riemannian gravity, it’s the group $SO(3)$ of rotations in 3D. Actually, I lied: each of these has a double cover, and this is the gauge group (which allows for a spin structure. To simplify a lot of things in my thesis, I talk about the case where $G$ is some finite group, but eventually I’d like it to be $SU(2)$, the double cover of the rotation group $SO(3)$.

So we imagine the connection tells us how an observer would be rotated by the act of moving along a path. (There is a kind of trivialization of a bundle lurking behind this glib statement, but I’m putting that off). Now, some connections are physically the same, even though we describe them differently. They are related by gauge transformations, which are symmetries of the connections themselves. These amount to a way of changing the coordinate system in which we describe (say) our rotation: two rotations of 60 degrees around different axes are not “really” different, since the observer can turn one into the other by tilting her head. What’s traditionally done is to “mod out” by gauge transformations: take any two connections related in this way to be just the same, and throw away any information that distinguishes them. Instead, we can organize flat connections into a category – in fact, a groupoid – where the objects are the connections, and the morphisms are the gauge transformations. We can organize this into the category $hom(\Pi_1(M),G)$ of functors from the fundamental groupoid of a manifold into the gauge group (thought of as a one-object category).

What’s the point – from a physical point of view – of keeping all the extra structure of these morphisms? To make a long story short, they’re what ends up allowing the theory to classify particles as having spins, not just masses. (Incidentally, I notice that Marni Sheppeard made a guest post on another blog arguing that category theory is useful to physics. Here is another example of how this can be so. Morphisms encode information that would be absent without them, and which has a straightforward physical meaning.)

How does this extra information appear? Well, first of all, what is a point particle, in this model? It’s represented as a boundary around a puncture in “space” – a circular boundary in a 2D surface of some shape or other. The fundamental groupoid of the circle has objects which are points of the circle, and morphisms which are (homotopy classes of) paths. There is an equivalence of categories between this and the fundmental group of the circle, which we can think of as a category with just one object (this is because the circle is a connected space).

Then we’re looking at a category $hom(\pi_1(S),G)$ of functors between a couple of one-object categories. Since $\pi_1(S) \cong \mathbf{Z}$, these are determined by the image of the generating path, “1”. So the groupoid of flat connections on this boundary has objects which correpond just to elements of $G$. But wait! There’s more! You also get natural transformations between these functors! These amount to just conjugations relating elements of $G$ (those “coordinate transformations” I mentioned before). So the whole groupoid has objects corresponding to elements of $G$, and morphisms $h: g \rightarrow g'$ for each $h$ such that $g' = h g h^{-1}$. We call this whole groupoid by the name $G /\!\!/ Ad(G)$ – or “$G$ weakly modulo the adjoint action of $G$.

This is also equivalent (as a category) to a smaller category I’ll call $skel( G /\!\!/ Ad(G) )$ – the “skeleton” of $G /\!\!/ Ad(G)$, namely, a category with one object for each isomorphism class of objects in $G /\!\!/ Ad(G)$ (i.e. each conjugacy class in $G$). Each of these has a group (the original category was a groupoid, so the new one is also) of automorphisms. This will be the same as the group of automorphisms of the corresponding object in $G /\!\!/ Ad(G)$ – namely, the stabilizer subgroup of that element of $G$, which, if $G = SU(2)$ is generically $U(1)$, except for a couple of exceptional points corresponding to 0-degree and 360-degree rotations.

Finally, a 2-vector in the 2-vector space assigned to the circle (which I like to think of as a “2-state”) is a functor from this $skel (G /\!\!/ Ad(G))$ into $\mathbf{Vect}$. Each such functor $F$ is a direct sum of a bunch of irreducible ones, and the irreducible ones assign a nontrivial vector space $F(g)$ to just one object $g \in skel (G /\!\!/ Ad(G))$ – and the group of automorphisms of that object are taken to a group of automorphisms of $F(g)$. That is, $F$ is specified by a conjugacy class of $G$, and a representation of its stablizer subgroup. If $G = SU(2)$, this is an angle and a spin. And in 3D gravity, the mass of a particle corresponds to an angle, because Einstein’s equation here says that space is locally flat, except where there is matter – where there is an amount of curvature proportional to the mass. This shows up as an “angle deficit” – an amount by which you end up rotated if you travel around the particle.

So that’s how you can see a “hole” in “space” as a point particle with mass and spin in this kind of extended TQFT. In higher dimensions, something similar happens, but the classification is more complicated, because in general the matter looks like “stringy” loops (this is something Derek Wise has looked at in his thesis). Also, above 3D, a theory of flat connections is no longer a theory of gravity, but rather something called BF theory – although in 4D it happens to be a limit of the theory of gravity as you allow Newton’s constant to approach zero. (That is, it describes the topological sector of the theory of gravity.)

What I haven’t yet explained is how this matter, which so far has the properties we might hope for, also gets to live in a spacetime governed by the Ponzano-Regge model. That means looking at what the extended TQFT does to the morphisms and 2-morphisms of the cobordism category – to “space” and to “spacetime”, and what the “2-linear maps” and “transformations” they give are like. Tune in next installment…