I’m going to be giving a talk on extended TQFT stuff and quantum gravity at Perimeter Institute next thursday, and then in mid-March I’ll be heading to UC Davis to give the same/similar talk for the String Theory and Quantum Gravity seminar being run by Derek Wise. So I have a bunch of things on my mind right now. However, before heading to Davis, I wanted to go back and look at some of the stuff Derek has done having to do with Cartan geometry, which I was following somewhat at the time, and blog about it a bit here. Before that, I’d like to wrap up this presentation of the talks I gave here about representation theory of the Poincaré 2-group, $\mathbf{Poinc}$.

As a side note, thanks to Dan for pointing out these notes on representations of the (normal, uncategorified) Poincaré group, including some general comments on representations of semidirect products. It’s interesting to consider how this relates to the more general picture of 2-group representations – but I won’t do so here and now.

In Part 1 I talked about what representations 2-categories of 2-groups are like in general, and in Part 2 a fairly concrete description of $\mathbf{Poinc}$. Here I’ll wrap up by summarizing the results of Crane and Sheppeard about what $Rep(\mathbf{Poinc})$ looks like concretely.

It has three parts: the objects are representations (also known as functors from $\mathbf{Poinc}$ as a 2-category with one object, into $\mathbf{Meas}$); the morphisms are 1-intertwiners (a.k.a. natural transformations) between reps; and the 2-morphisms are 2-intertwiners (a.k.a. modifications) between 1-intertwiners.

1) Representations: A functor

$\mathbf{Poinc} \rightarrow \mathbf{Meas}$

will pick out some measurable space $X = F(\star)$ for the lone object of the 2-group – or rather, $Meas(X)$, the 2-vector space of all measurable fields of Hilbert spaces on $X$. (This is a matter of taste since to know the one is to know the other.) Then for the morphisms and 2-morphisms of $\mathbf{Poinc}$ we get, respectively, 2-linear maps from $Meas(X)$ to itself, and natural transformations between them.

The morphisms of $\mathbf{Poinc}$ are just the group $G$ in the crossed-module picture I described in Part 2. For the usual Poincaré 2-group, this is $SO(p,q)$. For each such element, we’re supposed to get an invertible 2-linear map from $Meas(X)$ to itself – that is, a measurable field of Hilbert spaces on $X \times X$ (together with measures to do “matrix multiplication” with by direct integrals). This can only be invertible if the only Hilbert spaces which appear are 1-dimensional (since these maps compose by a “matrix multiplication” involving direct sums of tensor products of the components – and the discreteness of dimensions means that if any dimension is higher than 1, you’ll never get back the identity).

So any representation turns out to give what amounts to an action of $SO(p,q)$ on $X$ – the component $F(g)(x_1,x_2)$ is $\mathbb{C}$ if $x_2 = g \triangleright x_1$ and 0 otherwise. An irreducible representation gives an $X$ with a transitive action (otherwise, you can decompose it into orbits, each of which corresponds to a subrepresentation). Crane and Sheppeard classify several kinds of these, associated to various subgroups of $SO(p,q)$, but an easy example would be a mass shell in Minkowski space – a sphere or hyperboloid (depending on $(p,q)$) that is the full orbit of some point under rotations and boosts (a “mass shell” because it gives all the possible momenta for a particle of a given mass, as seen by an observer in some inertial frame).

The 2-morphism part of $\mathbf{Poinc}$ gives a homomorphism from $\mathbb{R}^{p+q} \rightarrow Mat_1(\mathbb{C})$ at each of these points. Now, one-by-one matrices of complex numbers are just complex numbers, so what we have here is a character of $\mathbb{R}^{p+q}$ – at each point on $X$. To be functorial, this has to be done in an equivariant way (so that acting on the point $x \in X$ by $g \in SO(p,q)$ affects the character by acting on $\mathbb{R}^{p+q}$ by the same $g$).

2) 1-Intertwiners:

If representations $F$ and $F'$ correspond to actions of $SO(p,q)$ on spaces $X$ and $X'$ respectively, with characters $h, h'$, then what is a 1-intertwiner $\phi : F \rightarrow F'$? Remember from Part 1 that it’s a natural transformation: to the object $\star$ of $\mathbf{Poinc}$ it assigns a specific 2-linear map

$\phi(\star) : F(\star) \rightarrow F'(\star)$

To each $g \in SO(p,q)$ (object of $\mathbf{Poinc})$ it gives a transformation

$\phi(g) : \phi(\star) \circ F(g) \rightarrow F'(g) \circ \phi(\star)$

This is a specified map which replaces the naturality square in the old definition of an intertwiner. It has to make a certain “pillow” diagram commute (Part 1).

Now, back in the posts on 2-Hilbert spaces, I explained that a 2-linear map $\phi(\star)$ is given by some field of Hilbert spaces $\mathcal{K}$ on $X \times X'$ (a “matrix” of Hilbert spaces, though of course $X, X'$ needn’t be finite), along with a family of measures on $X$ indexed by $X'$ (which allow us to do integration when doing the sum in “matrix multiplication”). The transformations $\phi(g)$ also can be written in components, so that

$\phi(g)_{(x,y)} : \mathcal{K}_{(F(g)^{-1}(x),y)}\rightarrow \mathcal{K}_{(x,F'(g)(y))}$

(Note this uses the two actions given by $F,F'$ on $X,X'$ – one forward, and one backward. This is the current form of what, in uncategorified representation theory, would be a naturality condition.)

What does this all amount to? One way to think of it is as a representation of $SO(p,q) \ltimes R^{p+q}$ itself! In particular, it’s a representation on the direct sum of all the Hilbert spaces which appear as components of $\phi(\star)$. This is since the maps given by the $\phi(g)$ have to satisfy a condition which says that composition is preserved (as long as you’re careful about indexing things):

$\phi(gg')_{(x,y)} = \phi(g)_{F(g')x,G(g')y)} \circ \phi(g')_{(x,y)}$

To get a representation of the group, we can say that elements $(g,h) \in G$ shuffle vector spaces over points in $X$ by the action of $g$ and then act within vector spaces by $h$. So then $\phi$ has both intertwiner-like and representation-like properties.

The “intertwiner-ness” of $\phi$ has to do with how it interpolates between two actions on $X,X'$ by turning them into an action on the product $X \times X'$ – but it also has some “representation-ness”, by giving this action of a (semidirect product) group on a big vector space.

3) 2-intertwiners

If a 1-intertwiner can be thought of as a representation of $G \ltimes H$, it shouldn’t be too surprising that a 2-intertwiner between 1-intertwiners $\phi, \phi'$ ends up being an intertwiner between the associated representations. If 1-intertwiners have some qualities of both reps and intertwiners, the 2-intertwiners are more single-minded.

In particular, a 2-intertwiner $m : \phi \rightarrow \phi'$ assigns to the only object of $\mathbf{Poinc}$ a 2-morphism in $\mathbf{2Vect}$ (that is, a field of linear maps between the vector spaces which are the components of $\phi, \phi'$), which satisfies some “pillow” diagram. When we form the big rep. by taking a direct integral of all those spaces, the field of linear maps turns into one big linear map, and the diagram it satisfies just collapses into the condition that it be an intertwiner.

So the representation theory of this interesting 2-group looks a lot like the representation theory of the group of 2-morphisms. The extra structure involving actions on measurable spaces by $G = SO(p,q)$ would be mostly invisible if you just thought about irreducible reps of the group, since the space would be just a single point.

This phenomenon where a lower-order structure turns up in some form at the top level of morphisms of its categorified version has cropped up before in this blog – namely, when extended TQFT’s turn out to contain normal TQFT’s in individual components. In these examples, categorification is less a matter of building more floors “on top” of structures we already know, as “higher morphisms” suggests, but excavating additional floors of subbasement – interpreting what were objects as morphisms.