## May 20, 2008

### Relation between AQFT and Extended Functorial QFT

#### Posted by Urs Schreiber

Update: the article is now on the arXiv.

This is your last chance (your first chance was here) to make it into the acknowledgements of

Urs Schreiber
On the relation between algebraic QFT and extended functorial QFT
arXiv:0806.1079

by complaining about which important references I missed, or complaining about how un-understandable the main argument is, or other complaints like this.

Abstract. There are essentially two different approaches to the axiomatization of quantum field theory (QFT): algebraic QFT, going back to Haag and Kastler, and functorial QFT, going back to Atiyah and Segal. More recently, based on ideas by Baez and Dolan, the latter is being refined to “extended” functorial QFT by Freed, Hopkins, Lurie and others. The first approach uses local nets of operator algebras which assign to each patch an algebra “of observables”, the latter uses $n$-functors which assign to each patch a “propagator of states”.

Here we present an observation about how these two axiom systems are naturally related: we demonstrate under mild assumptions that every 2-dimensional extended QFT 2-functor (“parallel surface transport”) naturally yields a local net. This is obtained by postcomposing the propagation 2-functor with the formation of 2-endomorphisms. The argument has a straightforward generalization to higher dimensions.

Posted at May 20, 2008 9:29 AM UTC

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### Re: Relation between AQFT and Extended Functorial QFT

There is now a sketch of an argument of how any equivariant structure on the QFT 2-functor induces a covariant structure on the correspondding local AQFT net. Section 6.

Posted by: Urs Schreiber on May 22, 2008 9:48 AM | Permalink | Reply to this

### Re: Relation between AQFT and Extended Functorial QFT

- the central computation showing that the pre-sheaf obtained from the the 2-transport is a local net, and that equivariance of the 2-functor implies covariance of the net is now entirely expressed in terms of 2-categorical pasting diagrams and hence much easier to visualize (“immediate proof by eyeballing the diagram”).

- I added the remark (easy demonstration) that the local net obtained from the 2-transport necessarily satisfies the time slice axiom.

- the example section has been expanded, contains now instructive lattice examples and 2-categorical traces.

- the appendix described 2-vector spaces and the canonical 2-representation which leads to FQFTs assigning von-Neumann algebras to points and bimodules to paths.

- last not least, the title has changed. Now it’s AQFT from $n$-functorial QFT.

You still have a few days to make it into the acknowledgements… :-)

Posted by: Urs Schreiber on May 27, 2008 9:59 AM | Permalink | Reply to this

### Re: Relation between AQFT and Extended Functorial QFT

Some typos at least:

‘We are displying a very symmetric configuration’ p. 15

In definition 7, A(g(O) is missing a bracket. p. 16

‘Bimod on all allgebras of the form…’ p. 27

Posted by: David Corfield on May 27, 2008 6:29 PM | Permalink | Reply to this

### Re: Relation between AQFT and Extended Functorial QFT

Thanks David! Am at Heathrow on my way to Stanford. Will incorporate this as soon as possible. Currently my wlan is defunct…

Posted by: Urs Schreiber on May 28, 2008 8:58 AM | Permalink | Reply to this

### Re: Relation between AQFT and Extended Functorial QFT

Say we take FQFT as one big exercise in the representation theory of cobordism categories. Did we decide whether all relevant types of cobordism fit into the generalized tangle hypothesis? Then what is there to say about how the algebra in the codomain of the representing functor receives that information about structure/stuff on the normal bundle? Is it all done through equivariance?

Posted by: David Corfield on May 28, 2008 9:37 AM | Permalink | Reply to this

### Re: Relation between AQFT and Extended Functorial QFT

Is it all done through equivariance?

Probably not. The crucial extra structure here for the phenomena that I am concentrating on in this article is pseudo-Riemannian metric structure on the cobordisms. And for what I am considering here, all the interesting information comes just from this extra structure, as topologically all cobordisms I am looking at are disks!

(The topological nontrivial aspects of the construction are supposed to be covered by ref [14]. Eventually that needs to be combined with the present discussion to yield one single unified description of 2-dimensional CFT in terms of 2-functorial transport.)

So this shares with the discussion of extended TQFT that we had the fact that it is “extended”, meaning $n$-functorial, but apart from that the presence of extra metric strucure radically changes the general picture we get. Which is not really a surprise.

Another thing I should add: there is a slight variation of the idea of extended functorial QFT which I think is important here:

from my point of view the natural assignment in extended FQFT is: you hand me any manifold on which you want to evaluate the eFQFT and what I hand you back is a differential cocycle on that manifold, in the form of an $n$-functor from $n$-paths in that manifold.

So that probes the entire manifold in an extended (localized down to points) way locally, but doesn’t a priori give an assignment of “holonomy”, namely of the result of “integrating” that $n$-functor over the entire manifold. Doing so will require performing higher traces. And these might not even exist in general.

So that’s the picture I have in mind:

FQFTs of Sigma-model type are:

start with an $n$-functorial differential cocycle on some target space $X$ (“classical parallel $n$-transport”).

then for any manifold (with boundary and corners, possibly) $\Sigma$ (the parameter space) quantization is some pull-push operation which produces from that classical differential cocycle on $X$ a corresponding differential cocycle on $\Sigma$, that’s the “quantum $n$-transport”.

finally, to get eQFT in the Baez-Dolan/Hopkins-Lurie sense: take traces and compute holonomies.

Posted by: Urs Schreiber on May 29, 2008 4:31 AM | Permalink | Reply to this

### Re: Relation between AQFT and Extended Functorial QFT

By private email somebody asked me how the construction of AQFTs from FQFTs that I give would generalize to the “Euclidean” case, where the pseudo-Riemannian metric is replaces with a Riemannian one.

I am not entirely sure. The construction I give makes crucial use of the Minkowski structure (or, more generally as indicated at the very end, of globally hyperbolic pseudo-Riemannian structure). I don’t see how to implement it directly with a Riemannian structure.

On the other hand, there are supposed to be general results on implementing Wick rotation in AQFT – Osterwalder-Schrader theory. But I am not sure to which extend that would help to relate 2-d Minkowski AQFT to for instance vertex operator algebras. There is something known about this – but I am not really sure what. :-)

Posted by: Urs Schreiber on May 29, 2008 5:21 AM | Permalink | Reply to this
Read the post HIM Trimester on Geometry and Physics, Week 4
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Excerpt: Talk in Stanford on nonabelian differential cohomology.
Tracked: May 29, 2008 11:28 PM
Weblog: Musings
Excerpt: A revived comment thread on an old post might be of interest to someone.
Tracked: June 1, 2008 5:00 PM

### Re: Relation between AQFT and Extended Functorial QFT

I have now expanded the (brief) discussion of chiral QFT (end of the example section 7) and boundary QFT (section 8.2), indicating how one obtains from FQFT the picture of boundary AQFT as described by Longo and Rehren.

Posted by: Urs Schreiber on June 1, 2008 7:07 PM | Permalink | Reply to this

### Re: Relation between AQFT and Extended Functorial QFT

I have now also added a bunch of references that are supposed to put everything in perspective:

I have included John’s “Quantum quandaries” as the best existing introduction to the idea that FQFT is about representations of cobordism categories. Then Huang/Kong on how that gives rise to VOAs.

And Abramsky-Coecke as the formalism making full use of this in Qm. Bob Coecke should emphasize this much more in his work: it’s not a mystery that string diagrams in monoidal categories are the language for quantum mechanics. But a consequence of the fact that QM is the representation theory of $1Cob_{Riem}$. So I added some references and remarks to this extent to help clarify this picture.

Posted by: Urs Schreiber on June 3, 2008 5:56 AM | Permalink | Reply to this

### Re: Relation between AQFT and Extended Functorial QFT

Hi Urs. I like this way of doing things, but I’ve never been able to see why every functor $1 \mathrm{Cob} _{\mathrm{Riemm}} \to Hilb$ should be of the form $t \mapsto e ^{Ht}$ for some operator $H$. It feels related to Stone’s theorem, but I can’t see how you can get anything like strong continuity from the category theory.

Also, the URL in the citation to Stolz and Teichner has an unescaped tilde which is being interpreted as an unbreakable space.

Posted by: Jamie Vicary on June 17, 2008 6:00 PM | Permalink | Reply to this

### Re: Relation between AQFT and Extended Functorial QFT

why every functor $1\mathrm{Cob}_{Riemm} \to Hilb$ should be of the form $t \mapsto e^{t H}$ for some operator $H$.

Right, not every functor does, but every smooth functor!

Heuristically:

$infinitesimal = smooth + functorial \,.$

Still hueristically, if something is functorial and also smooth, then the quantity it assigns to a large chunk is obtained from gluing the assignment to lots of tiny pieces. As you go to the limit, this says that the smooth functorial assignment comes from differential form data.

For the case at hand you make this precise, for instance, by using the identification of $1Cob_{Riemm}$ with paths in $\mathbb{R}$ and then use the result that smooth functors on paths with values in a group are equivalent to Lie-algebra valued forms.

This result is for instance in our article here.

There we keep talking about thin-homotopy classes of paths, since these form a groupoid.

It does not matter much for the case at hand, but in discussion with Stephan Stolz, Peter Teichner, it turned out that there is a certain desire to extend this result to just reparameterization classes of paths. I think if you go through the proof that Konrad and I present you’ll notice that it holds for just reparameterization classes of paths just as well, but unfortunately we didn’t state that generalization.

After a talk I gave Dan Freed pointed out that he states something like this result in his article on Classical Chern-Simons theory.

Posted by: Urs Schreiber on June 17, 2008 6:20 PM | Permalink | Reply to this

### Re: Relation between AQFT and Extended Functorial QFT

Also, the URL in the citation to Stolz and Teichner has an unescaped tilde which is being interpreted as an unbreakable space.

Oops. Thanks for letting me know!

Posted by: Urs Schreiber on June 17, 2008 6:27 PM | Permalink | Reply to this
Read the post A Groupoid Approach to Quantization
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### Re: Relation between AQFT and Extended Functorial QFT

I am receiving some feedback on my article by email and in personal conversation. It turns out that I should have emphasized the following in the example section:

a) I am not claiming that I have found the target 2-category such that 2-functors with values in it gives rise to local nets with values in type III von Neumann algebra factors under the construction I give.

b) At the same time, I think that such a 2-category should exist. In any case, I do not see why it should not exist.

c) In particular, I am not claiming that the example which I do discuss, that where the 2-functor takes values in the bicategory of vonNeumann algebras, their bimodules and bimodule homomorphism, leads to nets of type III factors. Clearly it does not. Instead, this example in the article only served the purposes of providing one further large class of local nets of algebras, namely one for each String-2-connection in 2-dimensions.

d) As all the other examples that I do discuss, all of them toy examples as far as real world QFT applications are concerned, these are examples of local nets but they are not of the type III kind which one hopes to see eventually.

e) While I do discuss the nets induced by constant 2-functors, I should have emphasized that an obvious way to get nets all of whose local algebras are isomorphic by is by starting with a weak 2-functor which assigns a weak identity 1-morphism to every path.

Posted by: Urs Schreiber on June 17, 2008 5:18 PM | Permalink | Reply to this
Read the post AQFT from Lattice Models (?)
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