In "Electric-Magnetic Duality and The Geometric Langlands Program", Sections 9 and 10, Kapustin and Witten describe certain convolution varieties in the affine Grassmannian (and more generally, in the Beilinson-Drinfeld) as moduli spaces of solutions to "the Bogomolny equations with 't Hooft operators added." While I can roughly make sense of what they are doing, it is not such easy reading for a mathematician, and of course, the proofs are pretty loose in nature. My (admittedly very vague) question is
Have any mathematicians followed up on this description i.e. written things in more mathematical language and done the proofs rigorously, or used it to understand the affine Grassmannian better?
Best Answer
Well, I think that there is no problem making that part of the paper rigorous (basically it is rigorous, modulo some well known results about moduli spaces of monopoles). In terms of how useful it is, the only thing that comes to my mind is this: it is a theorem of Jacob Lurie that the derived Satake category is an $E_3$-category, which means that you can make it live over the configuration space of points in a 3-dimensional space (informally $E_2$ is very close to just being symmetric monoidal and $E_3$ is some sort of higher commutativity; you can show that $E_3$ is the best thing you can hope for as the derived Satake category is not $E_4$ even for a torus). Now Lurie's argument is rather abstract, whereas probably you can give a purely geometric proof of this result using Witten-Kapustin construction (since they define some space over the configuration space of points in $\Sigma\times {\mathbb R}$ ($\Sigma$ is a Riemann surface) which simultaneously takes care of the "convolution" and "fusion" in the affine Grassmannian). This is not done anywhere but this is a well defined mathematical problem (define an $E_3$-structure on the derived Satake category using Witten-Kapustin space and show that it is equivalent to Lurie's).