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?