# The affine Grassmannian and the Bogomolny equations

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?

-
The paper definitely looks intimidating to me, as does the Math Reviews description by Siye Wu: MR2306566 (2008g:14018) Kapustin, Anton (1-CAIT-P); Witten, Edward (1-IASP-NS). Electric-magnetic duality and the geometric Langlands program. Commun. Number Theory Phys. 1 (2007), no. 1, 1–236. Have you followed up the dozens of citations listed in MathSciNet? (One is of course your own joint paper.) –  Jim Humphreys May 6 '11 at 21:50
The short answer is that I have looked at them, and none look promising. There's a problem with long papers like this: the point I'm asking about is a relatively minor part of a huge and influential paper, so the set of papers citing it has a high noise-to-signal ratio. –  Ben Webster May 6 '11 at 23:48
Incidentally, Google Scholar finds 250 citations, so there really are a somewhat overwhelming number to sort through. –  Ben Webster May 6 '11 at 23:49

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).

-
Does this theorem of Lurie on the derived Satake category have a reference? –  Peter McNamara May 12 '11 at 5:10
I am not sure - probably not at the moment. –  Alexander Braverman May 12 '11 at 5:29
I think you mean that $E_2$ is close to being braided. –  S. Carnahan Jun 10 '11 at 4:48

As far as I know the only paper which gives a mathematical version of those ideas is the following work of Benoit Charbonneau and Jacques Hurtubise.

http://front.math.ucdavis.edu/0812.0221

It doesn't quite use the language of the affine Grassmannian, but otherwise I think that it does what you want.

As far as understanding the affine Grassmannian better, the one application I know (which I already told you about) is to produce the symplectic structures on those slices that you (and Sasha, and I) like so much.

-