2-groups « Theoretical Atlas

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.

It’s been a while since I wrote the last entry, on representation theory of n-groups, partly because I’ve been polishing up a draft of a paper on a different subject. Now that I have it at a plateau where other people are looking at it, I’ll carry on with a more or less concrete description of the situation of a 2-group. For higher values of $n$ , describing things concretely would get very elaborate quite quickly, but interesting things already happen for $n=2$ . In particular, the case that I gave the talk about, a while back, was mostly the Poincaré 2-group, since this is the one Crane, Sheppeard, and Yetter talk about, and probably the one most interesting to physicists. It was first described by John Baez.

So what’s the Poincaré 2-group? To begin with, what’s a 2-group again?

I already said that a 2-group $\mathbb{G}$ is a 2-category with only one object, and all morphisms and 2-morphisms invertible. That’s all very good for summing up the representation theory of $\mathbb{G}$ as I described last time, but it’s sometimes more informative to describe the structure of $\mathbb{G}$ concretely. A good tool for doing this is a crossed module. (A lot more on 2-groups can be found in Baez and Lauda’s HDA V, and there are some more references and information in this page by Ronald Brown, who’s done a lot to popularize crossed modules).

A crossed module has two layers, which correspond to the morphisms and 2-morphisms of $\mathbb{G}$ . These can be represented as $(G,H,\triangleright, \partial)$ , where $G$ is the group of morphisms in $\mathbb{G}$ , $H$ consists of the 2-morphisms ending at the identity of $G$ (a group under horizontal composition).

There has to be an action $\triangleright : G \rightarrow End(H)$ of $G$ on $H$ (morphisms can be composed “horizontally” with 2-morphisms), and a map $\partial : H \rightarrow G$ (which picks out the source of the 2-morphism). The data $(G,H,\triangleright,\partial)$ have to fit together a certain way, which amounts to giving the axioms for a 2-category.

A handy way to remember the conditions is to realize that the action $\triangleright : G \rightarrow End(H)$ and the injection $\partial : H \rightarrow G$ give ways for elements of $G$ to act on each other and for elements of $H$ to act on each other. These amount to doing first $\triangleright$ and then $\partial$ or vice versa, and both of these must amount to conjugation. That is:

$\partial(g \triangleright h) = g (\partial h) g^{-1}$

and

$(\partial h_1) \triangleright h_2 = h_1 h_2 h_2^{-1}$

Both of these are simplified in the case that $\partial$ maps everything in $H$ to the identity of $G$ - in this case, $H$ can be interpreted as the group of 2-automorphisms of the identity 1-morphism of the sole object of $\mathbb{G}$ . In this case, by the Eckmann-Hilton argument (the clearest explanation of which that I know being the one in TWF Week 100) it turns out that $H$ has to be commutative, so the first condition is trivial since $\partial h = 1$ , and the second is trivial since it follows from commutativity. This simpler situation is known as an automorphic 2-group.

In any case, given a 2-group represented as a crossed module, automorphic or not, the collection of all morphisms can be seen as a group in itself - namely the semidirect product $G \ltimes H$ , which is to say $G \times H$ with the multiplication $(g_1,h_1) \cdot (g_2,h_2) = (g_1 g_2 , g_2 \triangleright h_1 h_2)$ . “What?” you may ask, or maybe “Why?”

Maybe a concrete example would help, since we’d like one anyway: the Poincaré 2-group, which is an automorphic 2-group. There are versions of various signatures $(p,q)$ , in which case $G = SO(p,q)$ , and $H = \mathbb{R}^{p+q}$ .

The group $G$ , then, consists of metric-preserving transformations of Minkowski space $R^{p+q}$ with the metric of signature $(p,q)$ - rotations and boosts (if any). The (abelian) group $H$ consists of translations of this space - in fact, being a vector space, it’s just a copy of it. Between them, they cover the basic types of transformation. Thinking of the translations as having a “projection” down to the identity rotation/boost may seem a bit artificial, except insofar as translations “don’t rotate” anything. More obvious is that rotations or boosts act on translations: the same translation can look differently in rotated/boosted coordinate systems - that is, to different observers.

So where does the Poincaré group $SO(p,q) \ltimes \mathbb{R}^{p+q}$ come in? It’s the group of all metric-preserving transformations of Minkowski space, and is built from these two types: but how?

Well, the vector space $H = \mathbb{R}^{p+q}$ is the group of transformations of the identity Lorentz transformation $1 \in G = SO(p,q)$ , since the map $\partial : H \rightarrow G$ is trivial. But suppose that there is another copy of $H$ over each point in $G$ . Then we have the set of points $G \times H$ , but notice that to talk about this as a group, we’d want a way to act on an element $h_1$ of one copy of $H$ over $g_1 \in G$ by another $h_2$ over $g_2$ . The obvious way is to just treat the whole set as a product of groups, but this misses the fundamental relation between $G$ and $H$ , which is that $G$ can act on $H$ , just as morphisms can act on 2-morphisms by “whiskering with the identity”. (Via Google books, here is the description of this in MacLane’s Categories for the Working Mathematician).

Concretely, this is the fact that there is a sensible way for both parts of $(g_1,h_1)$ to affect the $h_2$ , so we can say $(g_2,h_2) \cdot (g_1,h_1) = (g_2 g_1, g_1 h_2 + h_1)$ (using additive notation for translations, since they’re abelian). The point is that the first rotation we do, $g_1$ , changes coordinates, and therefore the definition of the translation $h_2$ .

So that’s the construction of the Poincaré group from the Poincaré 2-group. What would be nice would be to have some clear description of some higher analog of Minkowski space where it makes sense to say the Poincaré 2-group acts as a 2-group. I don’t quite know how to set this up, but if anyone has thoughts, it would be interesting to hear them.

One reason is that, when describing representations of the 2-group, there’s an important role for spaces (or at least sets) with an action of the group $G$ - which raises questions like whether there’s a role for 2-spaces with 2-group actions in representation theory of higher $n$ -groups. Again - I don’t really know the answer to this. However, in Part 3 I’ll describe concretely how this works for 2-groups, and particularly the Poincaré 2-group.

Theoretical Atlas

2-groups

n-group Representation Theory - part 3: Rep(Poinc)

n-group Representation Theory - (part 2: the Poincaré 2-group)

Pages

Blogroll

Archives

Categories