Witten Geometric Langlands Talk and Paper

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.

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105 Responses to Witten Geometric Langlands Talk and Paper

  1. Tony Smith says:

    Aaron and Peter, thanks for giving me more information. In light of it, I would revise my earlier statement from

    “If you try to extend the Fermi – Chew – Gross – Witten line, you see that the most prominent protege of Witten is Harvard Professor Lubos Motl.”

    to

    “If you try to extend the Fermi – Chew – Gross – Witten line, you see that the most prominent proteges of Witten are, as working physicists: Jon Bagger; Eva Silverstein; Shamit Kachru and Cumrun Vafa.
    In the eye of the general public, the best-known advocates of Witten’s conventional superstring theory are Brian Greene and Michio Kaku, but AFAIK neither of them come from the Fermi – Chew – Gross – Witten line, so that line cannot be credited (or blamed) for them.

    Thanks very much for the additional information. Now I am a bit less ignorant than before. (Aaron might say that I still have a long way to go, and he might be right, but at least I am making some progress.)

    Tony Smith
    http://www.valdostamuseum.org/hamsmith/

  2. Tony Smith says:

    Bert Schroer said “… Kurt Symanzik as well as Harry Lehmann and Wolfhart Zimmermann started their joint innovative work at Heisenberg’s institute, but scientifically they were kind of revolutionary “young Turks” in fierce opposition to [Heisenberg] who at that time was working at another of those ill-fated TOEs (Weltformel) in the veil of a nonlinear spinor theory. …”.

    Would it be fair to extend the Klein – Lindemann – Sommerfeld – Bethe, Pauli, Heisenberg, etc line
    to LSZ through Pauli, who called them the “Feldverein” ?

    As to Heisenberg’s TOE (Weltformel), didn’t Pauli disagree and withdraw, leaving the paper to be published by by Durr, Heisenberg, Mitter, Schlieder, and Yamazaki in Z. Naturforschg. 14a (1959) 441 ?
    Does Durr continue to work on it ?

    Tony Smith
    http://www.valdostamuseum.org/hamsmith/

  3. Bert Schroer says:

    Tony
    (with Peter’s permission), yes Pauli felt scientifically very close to the Feldverein. He certainly recommended Lehmann highly as a successor to Wilhelm Lenz. Pauli also felt a strong emotional attachment to the University of Hamburg where, at the recommendation of Sommerfeld (Lenz also came fro the Sommerfeld school) he got his first position as an assistant to Lenz (at the time when he had his epiphany about the exclusion principle while strolling along Rothenbaumchaussee). In Hamburg he also gave his last colloquium (about the neutrino). That’s when I saw the great man the first and last time. He also made his peace with Jordan but only after pouring all his sarcasm on him for having cultivated that naive Nazi sympathy (“Jordan is in the possession of a pocket spectrometer which allows him to distinguish between a deep red (communism) and an intense brown (the Nazi color)”) giving him the advice to care about his pension instead of meddling in politics.
    Sorry, let’s return again to the present where most of the interesting stories happen in the US.

  4. I can understand Laue was included in Farm Hall if only for affinity. Anecdotically, note he is buried next to Otto Hahn (If I remember well, they are right to Planck family burial, while Hilbert is left about thirty or forty steps).

    About Pauli, his Zurich lineage has produced Osterwalder and some other interesting researchers (eg some things from Albeverio or even from Frolich).

    I think lineages are interesting because one has always a tendence to follow the research track of his advisor, so if the advisor is well centered on productive physics his descendents are more likely to be (instead of random wandering around mathphys “interesting” problems). This explains, in part, the clustering of Nobel prizes along lineages.

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