There have been two quite interesting Oberwolfach workshops this summer with some relation to my favorite ideas about K-theory and quantum field theory. The most recent was a workshop on Gerbes, Twisted K-theory and Conformal Field Theory, with blogging from Urs Schreiber at The String Coffeee Table. Jouko Mickelsson gave a talk on “Twisted K-theory and the index on G” which from Urs’s description was mostly about the material in Mickelsson’s paper Families Index Theorem in Supersymmetric WZW Model and Twisted K-theory. This is closely related to the Freed-Hopkins-Teleman theorem, and their construction of a twisted K-theory class using Dirac operators on a circle, parametrized by connections on the circle.
Urs wasn’t sure what to make of this talk or how to connect it to string theory. My own point of view is that this is very interesting not because of the relation to strings, but because one can think of it as a possible new way of describing the Hilbert space for 2d chiral gauge theory. Perhaps this can provide a 2d toy model to test out new approaches to gauge theories in 3 and 4 dimensions. From this point of view, the QFT involved is best thought of not as the supersymmetric WZW model, but as a chiral fermion coupled to a gauge field, with BRST gauge fixing. In some sense what is going on here is an index-theoretic version of BRST.
Earlier in the summer there was an Oberwolfach workshop on Geometric Topology and Connections With Quantum Field Theory. One of the main topics there was recent work on elliptic cohomology, with a survey talk by Graeme Segal and Jacob Lurie speaking on a new “derived algebraic geometry” approach to the related theory of “topological modular forms”. Greg Moore’s talk looked interesting, especially his comments on various QFTs which he thinks of as special cases of AdS/CFT, and generalizations of the Chern-Simons/CFT correspondence. In a footnote he writes “It would constitute a major step forward in mathematics if someone could state the AdS/CFT correspondence in a mathematically precise way.”
The same Oberwolfach workshop also had a talk by Nitu Kitchloo on “The Baum-Connes Conjecture for Loop Groups”, which really was also about Freed-Hopkins-Teleman in disguise. I’ve talked a little bit with Paul Baum about this idea that FHT is Baum-Connes for loop groups, but Kitchloo has tried to do something with it. The general idea behind Baum-Connes is that one can study the representation theory of a group in terms of the topological K-theory of a classifying space for the group. In the case of loop groups, the classifying space is the space of connections on a trivial bundle over the circle, and the topological K-theory is FHT’s twisted K-theory of the group. The information about the loop group representation theory is encoded in the Verlinde algebra. An ongoing project of mine is to try and sort out the relations of this story to 2d QFT (see comment above about Mickelsson’s work), hoping that if one gets the right point of view on the 2d case one can use this to define gauge theories in 3 and 4 dimensions in terms of some sort of K-theory, implementing some sort of Baum-Connes correspondence for higher dimensional gauge groups.
How come whenever you talk about something with some mathematical substance, nobody ever comments on it, while they are happy to argue endlessly over X being rude to Y in Z’s blog?
Sigh….
I think there’s got to be a much slicker way to formulate and prove the relationship that the Freed-Hopkins-Teleman result addresses, but so far my best attempt is the paper I wrote with Alissa Crans, Urs Schreiber and Danny Stevenson relating the fundamental gerbe on a compact Lie group G to a 2-group (= categorified group) that one can also build from the central extension of the loop group of G.
I guess I need to go further and study how the representations of this central extension are connected to the K-theory of vector bundles on G twisted by the fundamental gerbe.
Hi John,
I’ll take a close look at the paper you mention, and see if you can convince me that I really need to think about Lie 2-algebras and 2-groups. My main interest in this is still QFT; I think there’s a beautiful QFT story going on here, one which I’ve only partly worked out, but have yet to see how the 2-group point of view helps.
You’re right that it’s kind of discouraging that my more substantial posts don’t get many comments. Partly my fault I think, for just referring to things without taking the time to write out some explanations of what is going on with this stuff (the kind of thing you do in TWF). This fall I hope to have more time for this and to get organized to be able to put formulas in these postings in some simple way. Maybe that will help.
I can’t really blame people though. The ongoing train wreck that is string theory and the bizarre behavior it is leading to is pretty fascinating to watch and hard to resist commenting on.