So one of the missing pieces in some of what I’ve been posting about recently is a discussion of 2-Hilbert spaces, and particularly the kind that categorify infinite dimensional Hilbert spaces. Part of the issue with these is that there are a number of ways of looking at them, and how these all fit together isn’t quite as clearly developed as with mere finite-dimensional 2-vector spaces.

I gave a little talk about this to our group at UWO, leading up to representation theory on 2-Hilbert spaces, which touches – potentially – on some of the stuff with spin foams that Wade and Igor are working on especially. It’s also a part of the project of trying to work out how the approach to extended TQFT’s I’ve described a bit should work for an infinite gauge group – in particular, a Lie group. The descriptions of these theories which I’ve given describe functors valued in $2Vect$, the 2-category of Kapranov-Voevodsky 2-vector spaces. These had a basis indexed by conjugacy classes in $G$, and representations of their stabilizer subgroups – and for, say, $G=SU(2)$, this is infinite.

Furthermore, a good categorification of a quantum field theory should use something deserving the name 2-Hilbert space (unless you prefer the $C^*$-algebra approach to quantum theory which doesn’t pick a specific representation on a Hilbert space – this would also be interesting to categorify). So it needs some structure analogous to an inner product, complex conjugation, and so no. There are a number of concepts that stab in this direction, so in my talk I tried to summarize the main points.

An early reference on this is HDA II by John Baez, which defines 2-Hilbert spaces in a nice axiomatic way – abelian categories, enriched over $Hilb$, with a *-structure, satisfying some properties, etc. There are a few provisos: the version of $Hilb$ things are presumed to be enriched over includes only finite dimensional Hilbert spaces. On the other hand, there’s no assumption – as there is for Kapranov-Voevodsky 2-vector spaces – that the category itself is finitely generated by simple objects. In other words, there’s no assumption of finite dimensionality for the 2-Hilbert space itself, but there is for its component Hilbert spaces. Then there’s a classification theorem which says what 2-Hilbert spaces in this sense are like. If they are finitely generated, then in particular they happen to be KV 2-vector spaces. But there is more structure, corresponding to two features of Hilbert spaces: the inner product, and complex conjugation.

The interesting thing about the inner product is that every KV 2-vector space is automatically equipped with one. Since it needs to be a map $\langle \cdot, \cdot \rangle : V^{op} \times V \rightarrow \mathbf{Vect}$, the obvious choice is $\langle V_1, V_2 \rangle = hom(V_1,V_2)$, which takes a pair of 2-vectors and gives a vector space – namely, the one containing all morphisms between these two objects. In $2Hilb$, the components of a 2-vector are themselves inner-product spaces, so we have a little extra structure. It turns out this has all the properties it needs to be a categorified inner product. As for the equivalent of complex conjugation, the categorified version is just adjunction – it leaves objects as they are, but turns morphisms into their (componentwise, vector-space) adjoints. This process has some important properties, such as being an involution (like conjugation) and so on. This makes $2Hilb$ into a *-category.

There’s another possible approach to the subject, or a closely related subject, is described by David Yetter in this paper on measurable categories, and which Crane and Yetter use to support representations of 2-groups, the way Hilbert spaces can support representations of groups. This is a more concrete, constructive approach – like describing $L^2$ spaces of complex functions on a topological space, rather than giving an axiomatic definition of a Hilbert space. Actually, what they describe is more like the space of measurable functions on a space. These are measurable fields of Hilbert spaces on a measurable space – such a field defines (a) a Hilbert space at each point, and (b) a space of “measurable sections”, namely ways of picking a vector in the space at each point which are considered measurable. (There are some properties, like the fact that the function giving the local norm of these vectors at each point is measurable, plus some closure-type properties.)

Well, that’s measurable. Given a measure, so you can do integration, you can define something like $L^2$ spaces. Integration works by means of the direct integal, which produces not a scalar, but a Hilbert space; in this kind of categorification, $Hilb$ takes the role of $\mathbb{C}$. The way this works is that the direct integral

$\int_X^\oplus F d\mu$

as a vector space is just the whole space of measurable sections. The inner product of sections is

$\langle f, g \rangle = \int_X \langle f_x, g_x \rangle_x d\mu$

So integrability of a measurable field means not finiteness, per se (which we think of as saying that an inner product gives a well defined map $\langle \cdot, \cdot \rangle : H^{\ast} \times H \rightarrow \mathbb{C}$), but that this direct integral gives an object of $Hilb$ (so the inner product integral should be finite, for instance, but also the space of measurable sections needs to be complete in the norm from this inner product). There is clearly a relationship between this way of describing an inner product and the way of describing it as a “hom-space”.

Some things are less clear… This gives a construction for how to get an “infinite dimensional 2-Hilbert space”. There doesn’t seem to be a known classification theorem here analogous to the one for KV 2-vector space, saying that this construction describes all “2-Hilbert spaces”. In fact, a general abstract definition of this concept seems to be a bit trickier than in the finite-dimensional case, and Crane and Yetter don’t really address it in their papers. One would hope that given a nice infinite-dimensional version of the usual definition of a 2-Hilbert space, this type would turn out to be generic.

Another question I’d like the answer to is – can one get one of these 2-Hilbert spaces from a (smooth, let’s say compact, probably) infinite groupoid, the way one can get 2-vector spaces (and, in particular, ones which can be made easily into finite-dimensional 2-Hilbert spaces) from an essentially finite groupoid? I think so – but there are some analysis issues to work out.  Assuming it works, this would be the right setting to support extended TQFT based on topological gauge theory with a Lie group like $SU(2)$ as gauge group.  (For some analysis reasons I may talk about later on, I only see reason to think this works with a compact group – but happily, that’s one right there!)

However, the question I’ll actually address in the second talk, which I’m giving on Friday, is how these are used for representation theory of 2-groups, since I’ve thought about that some, and some work has already been done with it – by, e.g. Crane, Yetter, Sheppeard, and also in some discussions I had the chance to participate in with John Baez, Laurent Freidel, Derek Wise, and Aristide Baratin (they are putting a paper together on the subject – as far as I know, not released yet).

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