Geometric Langlands at the KITP

There’s a very interesting program going on at the KITP discussing recent work of mathematical interest on 4d supersymmetric gauge theories (N=2 and N=4). These include various connections of 4d gauge theory to geometric Langlands uncovered by Witten and collaborators a few years ago, as well as last year’s conjecture by Alday-Gaiotto-Tachikawa of a relation between 4d gauge theory and 2d Liouville conformal field theory. In his introductory talk, Edward Frenkel discusses the possibility of a relationship between these ideas and the much earlier ideas about 2d conformal field theory that were inspirational at the beginnings of research on geometric Langlands (about which he has written extensively).

Yesterday and today Witten gave two talks on some new work. The first was about the very basic problem of how you quantize a finite dimensional symplectic manifold, which he approached using the phase-space path integral. The idea was similar to that described in a 2008 paper with Gukov, where the quantum mechanical problem gets turned into a 2d topological QFT problem. The innovation here is that he does this explicitly at the level of the path integral, using the kind of techniques for complexifying the problem, using holomorphicity and choosing appropriate path integral integration contours, that he pioneered in his recent paper on Analytic Continuation of Chern-Simons Theory. The second talk applied these ideas to the case of Chern-Simons theory. The path integral there is somewhat like the phase-space sort of path integral, and he expressed it in terms of a 4d QFT. He claims to be able to thus solve a well-known problem, that of how to get a QFT that gives Khovanov homology, which is a topological invariant with Euler characteristic the Jones polynomial. Unfortunately I get lost at the end when he has to go to 5 dimensions and perform some duality transformations. I gather he’ll have a paper about this relatively soon, and I’ll try again to see exactly how this works then.

Perhaps a collection should be taken up to buy a new camera for the KITP. The resolution of the one they have now been using for years is such that you often can’t quite read what the speaker is writing on the blackboard. Still, it’s wonderful to be able to follow along as they quickly put a lot of high-quality talks on-line.