2008 January « Theoretical Atlas

Recently I finished up my series of talks on 2-Hilbert spaces with a description of the basics of 2-group representation theory, and a little about the special case of the Poincaré 2-group. The main sources were a paper by Crane and Yetter describing 2-group representations in general, and another by Crane and Sheppeard. The Poincaré 2-group, so far as I know, was first explicitly mentioned by John Baez in the context of higher gauge theory. It’s an example of a kind of 2-group which can be cooked up from any group $G$ and abelian group $H$ , and which is related to the semidirect product $G \ltimes H$ .

One reason people are starting to take an interest in the representation theory of the Poincaré 2-group is that representations of the Poincaré group (among others) and intertwiners between them play a role in spin foam models for field theories such as BF theory, various models of quantum gravity, and so on. Some of these, turn up naturally when looking at TQFT’s, and generalizations of these, which is how I got here. Extending this to 2-groups gives a richer structure to work with. (Whether the extra richness is useful is another matter).

Before getting into more detail, I first would like to take a look at representation theory for groups from a categorical point of view, and then see what happens when we move to $n$ -groups - that is, when we categorify.

To begin with, we can think of a representation $(V, \rho)$ of a group $G$ as a functor. The group $G$ can be thought of as a category with one object and all morphisms invertible - so that the group elements are morphisms, and the group operation is composition. In this case, a representation of the group is just any functor:

$\rho : G \rightarrow Vect$

since this assigns some one vector space (the representation space, $\rho(\star) = V$ ) to the one object of $G$ , and a linear map $\rho(g): V \rightarrow V$ to each morphism of $G$ (i.e. to each group element) in a way consistent with composition. The nice thing about this point of view is that knowing a little category theory is enough to suggest one of the fundamental ideas of representation theory, namely intertwining operators (”intertwiners”). These are natural transformations between functors. This is the idea to categorify.

The point is that functors $F : G \rightarrow Vect$ can be organized into a structure $hom(G,Vect)$ , and this is most naturally seen as a category, not just a set. The category of representations of $G$ is usually called $Rep(G)$ , but seen as a category of functors, it is a general case of a category $hom(C,D)$ of functors from category $C$ to category $D$ . Let’s look at how this is structured, then consider what happens with higher dimensional categories. There seems to be a general pattern which one can just begin to see with 1-categories:

a functor is a map between categories, assigning
- to each $C$ -object a corresponding $D$ -object
- to each $C$ -morphism a corresponding $D$ -morphism
in a way compatible with composition and identities

a natural transformation between functors assigns
- to each $C$ -object a $D$ -morphism
making a naturality square commute for any morphism $g : x \rightarrow y$ in $C$ :

(In the case where the functors are representations of a group, this is an intertwiner - a linear map which commutes with the action of the group on $V$ .)

The pattern is a little more obvious for 2-categories:

a 2-functor is a map between 2-categories, assigning
- to each $C$ -object a corresponding $D$ -object
- to each $C$ -morphism a corresponding $D$ -morphism
- to each $C$ -2-morphism a corresponding $D$ -2-morphism
in a way compatible with composition and identities

a natural transformation between 2-functors assigns
- to each $C$ -object a $D$ -morphism
- to each $C$ -morphism a $D$ -2-morphism
making a generalized naturality square commute for any 2-morphism $h : f \rightarrow g$ in $C$ (where $f,g : x \rightarrow y$ ):

a modification (what I might have named a “2-natural transformation” or similar) between natural transformations assigns
- to each $C$ -object a $D$ -2-morphism
making a similar diagram commute (OK, well, it appears on p11 of John Baez’ Introduction to n-Categories, but I don’t have a web-ified version of it - I haven’t learned how to turn LaTeX diagrams into handy web format).

…

In the case where $C = G$ is a 2-group - a 2-category with one object and all $j$ -morphisms invertible, and $D = 2Vect$ , then we have here the (quite abstract!) definition of a representation, an 1-intertwiner between representations, and a 2-intertwiner between 1-intertwiners.

It’s not too hard to see the pattern suggested here - a “ $k$ -natural transformation” assigns a $k$ -morphism in $D$ to an object in the $n$ -category $C$ , and a $(k+j)$ -morphism in $D$ to each $j$ -morphism in $C$ . This morphism fits into a diagram filling a commutative diagram which was the coherence law for the top dimensional transformation for $(n-1)$ -categories. (I might point out that if I were to come up with terminology for these things from scratch, I’d try to build in some flexibility from the start. Instead of “functor”, “natural transformation”, and “modification”, I’d have used terms more analogous to the terminology for morphisms. Probably I’d have used, respectively, “1-functor”, “2-functor”, “3-functor”, and so on. This is already a problem, since these terms are in use with a different meaning! Instead, I’ve used “natural $k$ -transformation”.) It’s less easy to say what, explicitly, the various coherence laws should be at each stage, except that there should be an equation between the composites of (a) the $n$ -morphisms in an $n$ -natural transformation with (b) the two possible images of any chosen lower dimensional morphisms.

There is a lot of useful information out there about various forms of $n$ -categories, such as the Illustrated Guidebook by Cheng and Lauda, and Tom Leinster’s “Higher Operads, Higher Categories” (also in print). They’re a little less packed with information on functors, natural transformations, and their higher generalizations. I don’t know a reference that explains the generalization thoroughly, though. If anyone does know a good source on this, I’d like to hear about it. Probably this is somewhere in the work of Street, Kelly, maybe Batanin (whose definition of $n$ -category is the one implicitly used here) or others, but I’m not familiar enough with the literature to know where this is done.

These generalizations of functors and natural transformations to higher $n$ -categories describe what functor $n$ -categories are like. When written down and decoded, these definitions can be turned into a concrete definition of representations and the various $k$ -intertwiners involved in the representation theory of $n$ -groups.

However, next time I’ll take a look at some of what is known in the slightly more down to earth world where $n=2$ .

So I gave a little talk shortly before leaving London for Christmas. I had mostly written it up, but then I’ve been on the road for a while in Montreal, Ottawa, and Calgary, without consistent net access. However, now I have a moment to put this up.

The talk carried on from the previous one I described last post. It began to move in the direction of representation theory of 2-groups on 2-vector spaces and 2-Hilbert spaces, but didn’t get that far. This was partly because I had to finish describing what 2-linear maps and 2-maps look like for such spaces, and then because I had to explain about 2-groups and give some examples. I’ll say more about the representation-theory stuff in January. But here I’ll just summarize at least the rest of the description of the category $Meas$ , and also $2Hilb$ by describing 2-linear maps and so forth. Then I’ll comment a little more philosophically about what these are about.

So I explain how there’s a 2-vector space (in some suitable sense, not the KV sense) of measurable fields of Hilbert spaces on a space $X$ , analogous to the vector space of complex functions on a space. Also similarly, given a measure on $X$ , we get an inner product. Then there’s a (2-)Hilbert space where this inner product is always well-defined (as a complex scalar, or a genuine Hilbert space - which is the equivalent of a scalar at the next level up).

Well, then Crane and Yetter’s paper describes constructively how to get 2-linear maps (additive, linear functors) between such 2-vector spaces. They don’t as far as I can see, show that all functors arise this way, but it seems likely. The way is to say you get a functor $T: Meas(X) \rightarrow Meas(Y)$ from:

1) A measurable field of Hilbert spaces $T \in Meas(X \times Y)$ (this is similar to the linear maps between KV 2-vector spaces, which are like matrices of vector spaces)

2) A $Y$ -indexed family of measures $d \mu_y (x)$ on $X$ - these give you the measures you need to do the “inner product” involved in “matrix multiplication” at each $y \in Y$ (note that this stuff is only well-defined up to sets of measure 0, as usual). So we have, on objects:

$(T \mathcal{H})_y = \int^{\oplus}_X d \mu_y(x) T_{(x,y)} \otimes \mathcal{H}_x$

and a related expression for morphisms, using the identity on $T_{(x,y)}$ .

It’s probably worth pointing out that the measures on $X$ are used in the direct integral here, and so their only real role is to define the inner product on $(T\mathcal{H})_y$ - the underlying vector space at each point in the new field would be the same no matter what these measures were (up to the fact that if the resulting inner product is degenerate, we need a quotient space where it’s not).

So this gives 2-linear maps, which are functors. Natural transformation between these functors come from the fact that $Meas(X \times Y)$ is itself a category, and in fact a 2-vector space in the sense we’re using here ( $Meas$ is “enriched over itself”). So morphisms between these fields of Hilbert spaces basically amount to fields of bounded operators as usual. This is actually not quite right, because we need to account for the different measures: basically, you use a measure which is the geometric mean of those associated to source and target - check out Crane and Yetter’s paper if you want the details.

That finishes up a summary of how 2-Hilbert spaces work. The next thing I’ll be talking to our group about is how to use these for a categorified form of representation theory.

…

But first, what is the point of all this stuff? Not yet asking about representation theory in this setting - why is it interesting enough to bother? It’s worth thinking about what a categorification of a Hilbert space is supposed to be. In particular, let’s try locating them in the world of quantum mechanics.

A quantum system is usally portrayed as having states represented by vectors in a Hilbert space. The only things you can “do” to states involve applying operators to the whole space: project them into subspaces, “rotating” them by some unitary evolution operator, and so on. In a 2-Hilbert space, states, or “2-vectors” are objects in a category, which means there are not only these “macro” operations on the whole space, but also morphisms between any two states you pick. In fact, this is the source of the inner product on a 2-Hilbert space - there is a Hilbert space (in the usual sense) of morphisms between any two states, and in the world of 2-Hilbert spaces, this is the equivalent of a scalar.

In QM, the inner product $\langle x , y \rangle$ is telling you an amplitude to observe a system in state $y$ if it was set up in state $x$ - this is saying something about “how related” $x$ is to $y$ . The categorifed picture saying this is just $hom(x,y)$ makes more explicit what kind of relationship this is.

Now, if you happen to pick the same vector to start and end with, considering $\langle x , x \rangle = hom(x,x)$ , what this is saying is that there’s some bunch of “symmetry operations” on a state. (Taking just the invertible ones gives an actual symmetry group for a given state.) This is saying that “state 2-vectors” have some internal degrees of freedom. Their amplitudes give a measure of how many such degrees of freedom there are.

The fact that a 2-Hilbert space is described as an enriched category means that the usual picture of a quantum system returns when you look in individual components of a state 2-vector. In particular, the coefficients of a 2-state vector can be thought of as Hilbert spaces representing a system in that particular component. So, for instance, part of the big project I’m describing in these notes is to depict quantum gravity (at least in 3 dimensions) as an extended TQFT, which represents a physical system with these 2-Hilbert spaces. A 2-state vector here describes the situation on a boundary of space - matrix elements of a 2-linear map are Hilbert spaces of connections on a given manifold interpolating between chosen boundary states. Natural transformations between 2-linear maps are what give amplitudes for spacetimes joining such slices of space.

So what is a state 2-vector? All these properties should fit together into some nice scheme: classical configurations can exist in a “2-state” in some kind of superposition, where each configuration gets its own internal degrees of freedom. The inner product emerges naturally from this, considering morphisms between 2-states. Every morphism between 2-states has to respect the classical configurations, giving for each one a map between the internal spaces associated to it in the two 2-states. Is there a more elegant way to sum this up? Probably so, but at the moment I don’t quite see how to put it.

However: next time, I’ll carry on with some representation theory.

Theoretical Atlas

January 2008

n-Group Representation Theory - Part 1

2-Hilbert Spaces - pt 2

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