Geometric Langlands on the Beach

I’ve written a bit about the Geometric Langlands Program and its relation to physics here late last year, confessing to being confused about what it was supposed to have to do with N=4 supersymmetric Yang-Mills. Yesterday Witten gave a talk on the beach at the Simons workshop going on at Stony Brook. I’ve just finished listening to it, and it clarified things quite a bit for me.

Only having audio and no video is a bit frustrating, since not all the details of the equations get spoken, so sometimes you have to guess what the equation really is. In this case it’s a bit charming since you get to listen to the seagulls, waves and kids playing on the beach in the background. Maybe at some point lecture notes will be posted, and presumably Witten is writing up a paper on this material that will appear sooner or later, at which point I’ll try to get a better understanding of the details of this.

The idea seems to be to use a TQFT given by a twisted version of N=4 supersymmetric Yang-Mills, a slightly different one than the one studied by Vafa and Witten back in 1994. Then one does dimensional reduction using as 4-manifold a Riemann surface times the upper-half-plane, and ends up with a sigma model of maps from the upper-half-plane to the Hitchin moduli space of flat connections on the Riemann surface. The boundary degrees of freedom are branes, and the S-duality of the 4-d theory is supposed to give a duality at the level of the sigma model that corresponds to the fundamental duality one is trying to understand in the geometric Langlands program. The Hecke eigensheaves studied by mathematicians in this language are related to “magnetic eigenbranes”. Witten makes use of Wilson and ‘t Hooft operators studied in this context by Kapustin, and also mentions some related purely mathematical results of my colleague Michael Thaddeus and his collaborator Tamas Hausel.

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