The n-Category Café

In the afternoon Ulrich Bunke reviewed for us the theory of representations of affine Lie algebras. I won’t try to reproduce my notes right now. Maybe later. But I am afraid I will be too lazy…

The other main event was the “gong show” with lots of very short participant talks.

Two things I found particularly interesting:

A)

Christoph Wockel talked about a new theorem which relates smooth to continuous principal bundles with possibly infinite-dimensional structure group.

He said that previous methods for finding smooth bundles $P'$ that are equivalent, at the level of topological (continuous) bundles, to a given topological bundle $P$ relied on techniques that made use of a smooth structure of the classifying space of the corresponding structure group. This has a couple of disadvantages, one being that it fails for many infinite dimensional groups, where a smooth structure on the classifying space is hard to come by. The notable exception he mentioned were groups of diffeomorphisms of manifolds.

He said that using a new method which, as far as I understood, worked entirely at the level of local transition data, Müller has shown last year that there is an equivalence $$G\mathrm{Bun}^\mathrm{top}(M) \simeq G\mathrm{Bun}^{\mathrm{smooth}}(M)$$ under (just) the assumption that the group $G$ is locally convex and that $M$ is locally compact.

This theorem applies in particular to $PU(H)$-bundles, which are one way to think about abelian bundle gerbes. Hence it does have applications to formulations of smooth twisted K-theory.

B) Chris Schommer-Pries, student of Peter Teichner gave a quick overview on his perspective on generalized cohomology and extended functorial quantum field theory.

In broad strokes this was reminiscent of the picture that M. Hopkins sketched: Hopkins Lecture on TFT: Infinity-Category Definition.

I learned that a cool way to think about generalized cohomology which is alternative to infinite loop spaces is that related to diagram spectra.

Need to lear about these. Maybe I could even try to trick somebody knowledgeable reading this here into helping out with a couple of hints.

Then, concerning my efforts of understanding how to connect AQFT with extended functorial QFT I found help by a couple of experts who disabused me of various misconceptions and offered various very helpful hints.

There was some discussion about whether or not – and if then how – AQFT would know about the higher categorical structures that are suggested by extended functorial QFT. We could not pinpoint any hard statements, but it was pointed out to me that John Roberts, based on his old (and pioneering) ideas on nonabelian cohomology kind of conjectures that there is, somehow, a second nonabelian cohomology (somehow taking values in von Neumann algebras) associated with a net of observables of a gauge field theory, which would go beyond what is considered to date, and which would somehow encode crucial information about observables of gauge theories.

I got only very vague information on this particular aspect, but it was suggested that I take a look at the firsy part of an article by Roberts called Lectures in AQFT, which appeared in the book “Theory of superselection sectors”, edited by Kastler.

Similar (but more recent) information can apparently be found in the CIME Lectures on Noncommutative Geometry . I’ll try to get hold of these when there is time.

Finally, it was pointed out to me that a rich source of ideas about how to relate AQFT to the rest of the world is

Roberto Longo
Notes for a Quantum Index Theorem
math.oa/0003082.