Quantum Field Theory, As Seen By Mathematicians

In recent years, interest in quantum field theory among mathematicians has gone through ups and downs, in a sub-field dependent manner, as ideas rooted in quantum field theory have turned out to be mathematically useful in a variety of contexts. At Berkeley, it appears that there’s now more interest in quantum field theory in the math department than in the physics department, so much so that last fall three senior faculty (Nicolai Reshetikhin, Peter Teichner and Richard Borcherds) offered courses on various QFT topics. Luckily, Berkeley graduate students seem to be very industrious, and they have accumulated quite a few tex-ed lecture notes from these and other courses, gathering together the QFT ones here and here.

Another popular topic at Berkeley has been geometric Langlands, a subject where Witten’s QFT approach has intrigued many mathematicians. Witten has a new preprint aimed at mathematicians promoting the QFT point of view, and he explains in more detail claims he made in talks last fall that the use of 4d QFT corresponds in a sense to the use by mathematicians of “stacks”. Stacks are a mathematical device useful for handling non-free quotients, situations where one wants to keep track not just of the quotient space (which is often singular), but of more structure, for instance the varying stabilizer groups at different points of the quotient. Witten notes that if one just uses mirror symmetry of the Hitchin moduli space to study Langlands duality, one doesn’t know how to handle various singularities. Mathematicians have dealt with these singularities by invoking stacks, Witten instead argues that one should use a 4d gauge theory QFT perspective to see how to study the issue in a way that does not involve these singularities (they only appear when you dimensionally reduce and work with the 2d topological sigma models).

Last week Witten gave a talk to the mathematicians at the IAS on “Duality from Six Dimensions”, which is to be continued this week. He explained how the existence of a 6d superconformal theory implies SL(2,Z) symmetry and thus duality (Montonen-Olive duality) in the 4d N=4 supersymmetric topological gauge theory he uses in his approach to geometric Langlands. This is an old story by now, from the mid-nineties duality days, and Witten wrote up some of it here, for his contribution to the proceedings of the conference in honor of Graeme Segal’s sixtieth birthday back in 2002. David Ben-Zvi was at the talk taking notes, and I hope he’ll be adding to his extensive collection of on-line notes this semester since he’ll be at the Institute attending this year’s program there.

Note added: David has started posting his notes, the notes from the Witten talk are here.

At MIT this semester there’s a “pre-Talbot” seminar being run that will lead up to a Talbot workshop in March. The topic is something that might be called “quantum geometric Langlands”, involving not Witten’s QFT ideas, but a version of geometric Langlands that uses quantum groups due to Dennis Gaitsgory and Jacob Lurie. Scott Carnahan discusses this at Secret Blogging Seminar, and has notes from his overview talk available.

In March Lurie will be giving the Marston Morse lectures at the IAS, on the topic of “Topological Quantum Field Theories in Low Dimensions”.

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7 Responses to Quantum Field Theory, As Seen By Mathematicians

  1. David Ben-Zvi says:

    Hi Peter — yes I’m taking notes and will start posting them
    very soon (under a heading as notes from
    this semester’s activities)

    On another note the quantum geometric Langlands
    program fits very nicely with the Kapustin-Witten
    TFT point of view — the “quantum” parameter
    here is exactly the parameter Psi that appears
    in the 4d TFT (which is a combination of the gauge
    coupling of N=4 superYangMills and the parameter
    that’s used in defining the topological twist).
    The idea has been around on the math side
    for a while but Dennis and Jacob have formulated it much
    more precisely (and given a “local” version), and are
    explaining its relation to quantum groups.. it’s worth pointing out
    that the same parameter appears as the level of a Kac-Moody
    algebra or (when a positive integer) as the level in
    Chern-Simons theory or (when exponentiated) as the q in
    quantum groups..

  2. A.J. says:

    Also worth noting: Peter Teichner is running another Hot Topics course this semester, this time on the work of Freed, Hopkins, & Teleman. I’m not sure if anyone is publishing notes this time, but I’ll definitely be blogging about the course from time to time.

  3. David Ben-Zvi says:

    Here’s the link to the IAS term notes (I’ll add to it
    as the term progresses):

    http://www.math.utexas.edu/users/benzvi/GRASP/lectures/IASterm.html

  4. Peter Woit says:

    Thanks David, both for the great public service of providing the notes, as well as for the indications of the the frightening degree to which these ideas are all interconnected…

  5. Voltberg says:

    QFT must be more important for mathematicians than it is now, especially for those in small departments, because it touches so many areas of mathematics. That forces to learn many developments in unconected areas.

  6. Florian says:

    Peter, thank you for this very informative post!

  7. Pingback: This past two weeks in the arXivs… « It’s Equal, but It’s Different…

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