At the KITP in Santa Barbara there’s a wonderful program on Gauge Theory and Langlands Duality starting up this week, with some of the talks beginning to become available. The main topic will be the relations between S-duality in quantum field theory and geometric Langlands duality that Witten and collaborators have been working on the past few years.
I started trying to watch the talks, but the fact that the video quality is such that one can almost but not quite tell what is being written on the blackboard makes this a bit of a trial. I’m hoping that David Ben-Zvi or someone else will make available notes, which would help a lot. I did very much like Edward Frenkel’s description of the Langlands story as a “Grand Unified Theory of Mathematics”, and was interested to hear that he still feels that there are two different stories about the relation to QFT here, whose relationship is not at all understood (S-duality in 4d QFT is one, 2d CFT and vertex algebras is the other). It seems that A.J. Tolland is there, maybe he or someone else will do some blogging. As I get time to take in the lectures, I hope to write some more about them here.
Update: Notes for the talks are now also being posted, making following them on-line much more feasible. The quality of the talks is excellent, with Ed Frenkel so far giving a beautiful introduction to the roots of the Langlands program in number theory, David Ben-Zvi explaining the structures in topological quantum field theory that mathematicians are trying to exploit, David Morrison and Paul Aspinwall explaining mirror symmetry, D-branes, and the relation to N=(2,2) superconformal field theory, with examples, and Anton Kapustin starting on the 4d N=4 TQFT used to turn S-duality into a mirror symmetry.