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.
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”.