The n-Category Café

Tom Lada and I are organizing a special session at the 2009 Spring Southeastern Section Meeting in Raleigh, NC, during April 4-5, 2009:

Thanks. I should have mentioned this. We had the announcement here.

But I haven’t yet found any funding to get me there…

You also missed the week when the world financial system collapsed!

I did have access to newspaper news. While they didn’t report on the latest development in extended TFT there for some reason, they did mention that financial collapse.

And thanks to globalization, part of the collapse is taking place here, too, if admittedly on a smaller scale.

Not that i want to steer you guys away from the more “practical matters”… but, a discussion about the last two papers you cited (on your post, namely Hollands’ and Witten’s) would really interest me

Thanks for taking interest. I’ll try to say something about Holland’s work first, soon.

Hollands proposes to axiomatize Euclidean quantum field theory in terms of the operator product expansion (OPE) of its poinlike fields. In fact, he goes one step further and suggests to take the OPE to be more basic than the concept of quantum field itself. So essentially his suggestion boils own to studying certain algebras which depend on parameters (spacetime points) and satisfy a refined associativity condition depending on these parameters (the OPE factorization property).

Of course essentially this is done one way or another commonly in chiral 2-dimensional and especially in chiral 2-dimensional conformal QFT, where this is essentially the picture of vertex operator algebras.

As we have recently mentioned a couple of times (first here), vertex operator algebras are now understood as precisely the algebras for the operad whose morphisms are conformal spheres with $n$ conformally parameterized incoming and one conformally parameterized outgoing insertion point which depend holomorphically on these insertion points. So VOAs have now a good interpretation in terms of Euclidean functorial QFT (“FQFT”).

In particular, the factorization property in this context is just the obvious functoriality, i.e the operad-algebra property, i.e. the respect for the compositition of spheres along their parameterized insertion points.

For that reason it seems to me that Hollands’ approach might benefit from a investigation into its relation to representations of the corresponding higher dimensional cobordism operads/categories.

Also, Hollands announces upcoming work with Wald on the generalization of their approach to the non-Euclidean Lorentzian setup. That should be more interesting, since it should involve in particular passing from chiral QFT to full QFT. From the hints he gives in the present article I came to wonder what he might have on top of the standard AQFT picture of nets of local operator algebras, which gives the OPE picture (in the Lorentzian case) by passing to “stalks” i.e. to field algebras living over regions asymptoting to points.

Actually recently, I sat in a talk Wald gave on this work with Hollands. But I didn’t pay much attention, because after the introduction of this talk I had the insight which has now become AQFT from FQFT # and I was busy the rest of the time drawing pictures on scratch paper… ;-)