So I recently received word that this paper had been accepted for publication by Applied Categorical Structures. Since I’ll shortly be putting out another which uses its main construction to build Extended Topological Quantum Field Theories, it’s nice and appropriate to say something about that. But actually, just at the moment, I want to take a slightly different approach.

Toward the end of February, I went up to Waterloo to the Perimeter Institute, where my friend Derek Wise was visiting with Andy Randono – apparently they’re working on a project together that has something to do with Cartan Geometry, which is a subject that plays a big role in Derek’s thesis.

However, Derek was speaking in their seminar about Extended TQFT (his slides are now up on his website, and there’s also a video of the talk available). Actually, a lot of what he was talking about was work of mine, since we’re working on a project together to constructs ETQFT’s from Lie groups (most likely compact ones at first, since all the usual analytical problems with noncompact groups turn up here). However, I really enjoyed seeing Derek talk about it, because he has a sharper grasp than I do of how this subject appears to physicists, and the way he presented this stuff is very different from the way I usually talk about it (you can see me in the video trying to help deal with a question at the end from Rafael Sorkin and Laurent Freidel, and taking a while to correctly understand what it was, partly because of this jargon gap – I hope to get better).

So, for example, describing a TQFT in the Atiyah/Segal axiomatic formulation is fairly natural to someone who works with category theory, but Derek motivated it as a way of taking a “deeper look at the partition function” for a certain field theory. The idea is that a partition function $Z$ for a quantum field theory associates a number to a space $M$, satisfying certain rules. It is usually described by some kind of integral. Typically in QFT, these are rather tricky integrals – a topological QFT has the nice feature that, since it has no local degrees of freedom, these integrals are much more tractable. Of course, this is a mathematically nice feature that comes at the expense of physical relevance, but such is life.

Anyway, the idea is that the partition function $Z$ for an $n$-dimensional TQFT can be thought of as assigning, not just numbers to $n$-dimensional manifolds $M$, but something more which reduces to this in a special case. Specifically, $Z$ assigns a Hilbert space to any codimension-1 submanifold of $M$, in a particular way which Derek passed over by saying it “satisfies some compatibility conditions”. For an audience of mathematicians, you can gloss over this just as quickly by saying the assignments are “functorial”, or even with more detail saying the conditions make $Z$ a symmetric monoidal functor.

Part of the point is that these conditions are about as obvious on physical grounds as they are if you’re a category theorist. For example, the fact that composition is preserved by the functor $Z$ can be interpreted physically as saying that the number $Z(M)$ given by the partition function isn’t affected by how we chop up the manifold $M$ to analyse it. The fact that $Z$ is a monoidal functor ends up meaning that the “unit” for manifolds under unions (namely, the empty manifold with no points, which you can add to things without affecting them) gets assigned the Hilbert space $\mathbb{C}$, which is the unit for Hilbert spaces with respect to the tensor product $\otimes$. The fact that this is so means we can treat a manifold with no boundary as going from one (empty) boundary to another (empty) boundary – it therefore gets assigned a linear map from $\mathbb{C}$ to $\mathbb{C}$ – a number. Seeing how this linear map comes from composing pieces of the manifold is what “a deeper look at the partition function” means.

ETQFT does essentially the same thing, at one level deeper. The point is that a TQFT breaks apart a manifold by treating it as a series of pieces – manifolds with boundary, glued together at their boundaries. An ETQFT does the same to these pieces, treating them as composed of pieces – manifolds with corners – which are glued orthogonally to the gluing just mentioned. That is, there are two kinds of composition, so we’re in some sort of 2-category (bi-, or double- depending on how you formulate things). The essential point is that now, to manifolds without boundary, which are of codimension 1, we assign Hilbert spaces – and to top-dimensional manifolds WITH boundary, we assign maps of Hilbert spaces.

An ETQFT attempts to give a “deeper-still look at the partition function” by seeing how the Hilbert space arises from composition of pieces in this new direction, along boundaries of codimension 2. The way Derek describes this for physicists is to say that the ETQFT describes how that Hilbert space is “built from local data”, which he described in the usual physics language of path integrals. First of all, the conventional thing in physics is to take $Z(\Sigma)$ for a (codimension-1) manifold $\Sigma$ to be $L^2(\mathcal{A}_0(\Sigma)/\mathcal{G}(\Sigma))$ – the space of square-integrable functions on the quotient of the space $\mathcal{A}_0(\Sigma)$ of flat $G$-connections on $M$ by the action of the group of gauge transformations $\mathcal{G}(\Sigma)$.

Given a manifold $M$ with boundary components $\Sigma$ and $\Sigma '$, the standard quantum field theory formalism to describe the map $Z(M) : Z(\Sigma) \rightarrow Z(\Sigma ')$ given by a TQFT is to describe how it interacts with particular state-vectors in the Hilbert spaces for the source and target boundary components of $M$. So then:

$\langle \psi | Z(M) | \phi \rangle = \int_{\mathcal{A}_0(M)/\mathcal{G}} \mathcal{D}A \overline{\psi(A|_{\Sigma '})} e^{i S([A])} \phi(A|_{\Sigma})$

The point being, a flat connection $A$ has some action on it, which depends only on its gauge equivalence class $[A]$ (“the Lagrangian has gauge symmetry”), and it restricts to give flat connections on $\Sigma$ and $\Sigma '$, on which the $L^2$-functions $\psi$ and $\phi$ act, to give something we can integrate. The measure $\mathcal{D}[A]$ is a crucial entity here, and in general can be a real puzzle, but at least for discrete groups, it’s just a weighted counting measure which effectively gives us the groupoid cardinality of the quotient space. As for the action $S$, the simplest possible case just says the action of any flat connection is zero – hence this expression is just finding the (groupoid) cardinality, or more generally measuring the (stacky) volume, of the configuration space for flat connections. There are other possible actions, though.

Derek gives an explanation of how to interpret this in terms of the “pull-push” construction, which I’ve talked about elsewhere here, including in the above paper, so right now, I’ll just pass to the next layer of the ETQFT layer cake – codimension-2. Here, there is a similar formula, which also has an interpretation in terms of a “pull-push” construction, but which can be written as a categorified path integral.

So now the $\Sigma$ has boundary, and connects “inner” codimension-2 boundary component $B_1$ to “outer” boundary component $B_2$. Then, say, $B_1$ gets assigned the category of all gauge-equivariant “bundles” of Hilbert spaces on $\mathcal{A}_0(B_1)$, rather than the space of gauge-invariant functions. (Derek carefully avoided using the term “category”, to stay physically motivated – and the term “bundle” is accurate in the case of a discrete gauge group $G$, but in general one has to appeal to the theory of measurable fields of Hilbert spaces, since they needn’t be locally trivial). Then given particular Hilbert bundles $\mathcal{H}$ and $\mathcal{K}$ on the spaces $\mathcal{A}_0(B_1)$ and $\mathcal{A}_0(B_2)$ respectively, we can define what $Z(\Sigma)$ is by:

$\langle \mathcal{K} | Z(M) | \mathcal{H} \rangle = \int_{\mathcal{A}_0(M)/\mathcal{G}} \mathcal{D}A \mathcal{K}(A|_{B_2}) \otimes T_A \otimes \mathcal{H}(A|_{B_1})$

The interpretation is much like the previous formula: now we’re direct-integrating Hilbert spaces, instead of integrating complex functions – and we get a Hilbert space instead of a complex number, but this is in some sense superficial. Something any physicist would notice right away (or anyone comparing this to the previous formula) is that the exponential of the action $S([A])$ seems to have gone missing, to be replaced by some Hilbert space $T_A$. If we’re using the trivial action $S \cong 0$, this is fine, but otherwise, how exactly $S$ affects the direct integral would take some explaining. For now, let’s just say that we should think of $S([A])$ as being folded into either the inner product on $T_A$, or into the measure $\mathcal{D}A$: it shows up in its effect on the inner product on the Hilbert space that this direct integral produces.

Let me jump to the end of Derek’s talk here, to get at some conceptual aspect of what’s happening here. The axiomatic way of talking about ETQFT, namely Ruth Lawrence’s way, is to say we assign a 2-Hilbert space to the codimension-2 manifolds. But “2-Hilbert space” is an off-putting bit of jargon, so instead the suggestion is to replace it with “von Neumann algebra”.

The point is that 2-Hilbert spaces are thought (according to a paper by Baez, Baratin, Friedel and Wise) to be just categories of representations of vN algebras. Being a 2-Hilbert space means, for instance, that they’re additive (by direct sum), $\mathbb{C}$-linear (there is a vector space of intertwiners between any two representations), have duals, and so on. Moreover, they’re monoidal 2-Hilbert spaces, since there is a tensor product. Their idea is that the two ideas correspond exactly. In any case, the way the ETQFT construction in question works actually passes through a von Neumann algebra. This comes from the groupoid algebra that’s associated to a certain group action. Namely, the action of the gauge group on the space of flat $G$-connections on the manifold $M$.

Then the way we can look more closely at the “structure of the partition function” is by seeing the Hilbert space associated to a codimension-1 manifold as actually being a kind of morphism of von Neumann algebras. In particular, it’s a Hilbert bimodule, which is acted on by the source algebra (say $A$) on the left, and the target algebra ($B$) on the right. This is intimately connected with the stuff I was writing about recently about Morita equivalence, and so to the 2-Hilbert space view. In particular, a Hilbert bimodule $H$ gives an adjoint pair of linear functors (or “2-linear maps”) between the representation categories of algebras.

So shortly I’ll make a post about some papers coming out, and get back to this point…