The Langlands Program and Quantum Field Theory

Edward Frenkel is here this semester in the math department at Columbia, and he’s giving a series of lectures on a topic dear to my heart. Video of his lectures on The Langlands Program and Quantum Field Theory is starting to be available, courtesy of our graduate students Alex Waldron and Ioan Filip, as well as our staff member Nathan Schweer.

The first lecture last week was an overview, outlining the general picture of the Langlands program in the number field, function field and geometric cases, as well as two sorts of connections to QFT (to certain 2d conformal field theories, and to S-duality in 4d super Yang-Mills). This week he started to get more specific, giving some details about how the Langlands program works in the function field case, in preparation for moving next week to the geometric analog where a curve over a finite field gets replaced by a Riemann surface. As an indication of references covering much of the material to be discussed in lectures, Frenkel suggests this survey article and this Seminaire Bourbaki report.

Frenkel is also working on some other different but quite interesting projects. With Ngo and Langlands he has a program to “geometrize” the trace formula, for details see his very recent AMS Colloquium Lectures. With Losev and Nekrasov he has a fascinating program for studying certain field theories using instantons in a very different limit than the usual semi-classical one. See here, here, and here.

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7 Responses to The Langlands Program and Quantum Field Theory

  1. Some Physics Bro says:

    Interesting links!

    I was reading through and on the 6th page of the first Frenkel, Losev, Nekrasov paper they rather off-handedly state (without reference) that “It is generally believed state that non-supersymmetric quantum field theories should be understood as non-supersymmetric phases of supersymmetric ones”. Is this a common piece of wisdom? I have never heard anything along these lines before. What does this even mean? Is there any sort of talk or paper that actually lays out an argument about this? Or is this just some personal conviction the authors decided to slip in without having to explain?

  2. Peter Woit says:

    SPB,

    You’re dropping the adjective “realistic” in that quote, it doesn’t sound like they are talking about a general phenomenon in QFT. I’m not sure what the authors had in mind, but one possibility is that they are just referring to the idea that the SM is really part of a SUSY theory with broken supersymmetry. This certainly has been a popular idea among theorists, perhaps less so after the past year of LHC results…

  3. Ossicle says:

    To what extent does his work intersect (or whatever verb you like) with you own ongoing efforts? Is his presence at Columbia this semester potentially fruitful for your work?

  4. Peter Woit says:

    Ossicle,

    There’s quite a lot of intersection of Frenkel’s work with what I’ve been trying to do, and having him here is great for me. We’ve already talked a bit, that has been very helpful and I’m sure this will continue. The next few weeks he’ll be covering in his lectures the topics closest to my interests (how 2d QFT gets used in geometric Langlands). The combination of the lectures and being able to talk to him about this should help clarify a lot for me and hopefully will lead to some significant progress (and to finally getting some of this written up…).

  5. Ossicle says:

    That’s terrific, Peter, good luck. I’m rooting for you to accomplish great things! :D

  6. Mike says:

    Peter,

    I was wondering if you’d seen the discussion at Sean Carroll’s blog about Lagrangian QFT.

    Here is the link:
    http://blogs.discovermagazine.com/cosmicvariance/2012/02/07/how-to-think-about-quantum-field-theory/#more-7957

  7. Peter Woit says:

    Mike,

    I wrote about this here:

    http://www.math.columbia.edu/~woit/wordpress/?p=4408

    To my mind it’s hard to overemphasize how poorly we understand QFT, and so “heuristic” as well as “rigorous” approaches are both worth pursuing.