I spent yesterday afternoon down in Princeton, and attended a talk by Witten at the Institute on his work relating gauge theory and the geometric Langlands program. He says that his paper with Kapustin is done, it’s about 220 pages long, and will appear on the arXiv in Monday’s listings. So Sunday night, this link to hep-th/0604151 should start working. He’s also working on a book on the subject, where he would be the sole author.
At the start of talk, Witten noticed that many of the Institute’s mathematicians were there, and warned them that they had come to the wrong talk, since it was one aimed at physicists. Pierre Deligne got up and left, but others, including Sarnak and Langlands himself, did stay for the whole thing, although I’m not sure how much they got out of it.
Witten began by giving an outline of the talk, emphasizing six main ideas that were crucial to what he wanted to explain. He also listed as number zero the idea of geometric Langlands itself, saying he would talk about it at the end if he had time (he didn’t). The six main ideas were:
1. From a certain twisting of N=4 supersymmetric Yang-Mills one can construct a family of 4d TQFTs parametrized by a sphere. The twisting is the same sort that occurs in his original TQFT for Donaldson theory, in that case coming from N=2 supersymmetric Yang-Mills. The TQFTs he considers have an S-duality, part of a larger SL(2,Z) symmetry.
2. Compactifying the theory on a Riemann surface leads to topological sigma models, based on maps from the Riemann surface into the Hitchin moduli space MH of stable Higgs bundles. The four dimensional S-duality corresponds here to a mirror symmetry of these topological sigma models.
3. Wilson and ‘t Hooft operators of the 4-d gauge theory act on the branes of the topological sigma models. Branes mapped in some sense to a multiple of themselves by these operators are called electric or magnetic “eigenbranes” respectively.
4. Electric eigenbranes correspond to representations of the fundamental group, this is one side of the geometric Langlands correspondence.
5. The ‘t Hooft operators of the gauge theory correspond to the Hecke operators of the geometric Langlands theory although these are now defined on the space of Higgs bundles, not G-bundles.
6. Using a certaing co-isotropic brane on MH, magnetic eigenbranes give D-modules on the moduli space of bundles. The electric-magnetic duality coming from S-duality in the gauge theory relates electric and magnetic eigenbranes, giving the geometric Langlands duality between representations of the fundamental group of the Riemann surface in the Langlands dual group, and “Hecke eigensheaves” on the moduli space of G-bundles.
By the time he got to the 6th of these ideas, he was running out of time and things got very sketchy.
Witten made clear that this work doesn’t directly give dramatic new physics or mathematics, but rather just explains some tantalizing relations between gauge theory and Langlands duality, ones that were first noticed in work of Goddard, Nuyts and Olive in 1976, and pointed out to him by Atiyah way back then. The geometric Langlands program is famous among mathematicians for its difficulty (I still have trouble getting my brain around the concept of a Hecke eigensheaf..), and for its tantalizing nature, bringing together a range of different mathematical ideas (many involving conformal field theory, although these seem to be different than what Witten is doing). The new relations between this subject and supersymmetric gauge theory and TQFTs that Witten has unearthed may very well lead to some very interesting new mathematical developments in the future. Undoubtedly it will take people a while to make their way through the new 220 page paper and absorb all that he and Kapustin have worked out since last summer.