The n-Category Café

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.

Recent additions:

- 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… :-)

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

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?

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. :-)

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.

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.

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.