TITLE: What are the applications of the Atiyah-Bott Yang Mills paper?
QUESTION [23 upvotes]: I recently finished a seminar going through Atiyah and Bott's paper ''The Yang-Mills Equations over Riemann surfaces''.  The ideas going into the proof were surprising and very beautiful to me.
However, beyond its proof's beauty, I'm having trouble seeing the use of what I've just read. For instance, as I understand it the main result of the paper is an inductive formula for the cohomology of the space $\mathcal{C}(n,k)$ (the holomorphic vector bundles of rank $n$ and Chern class $k$ over Riemann surface $M$). This makes what the $\mathcal{C}$ look like a little clearer to me, but I've heard that if $g(M)\ne 0,1$ no very explicit of the $\mathcal{C}$ are known, so the only application I can think of (helping obtain an explicit description of the $\mathcal{C}$) seems not to have worked yet.
That naive train of thought lead me to ask:

What subsequent mathematics has heavily used the results of the
  Atiyah-Bott paper? Or, more petulantly, what's the point of the
  result?

(I know that there was a lot of activity on the Yang-Mills ideas which appear in the proof by Donaldson etc., but I'm asking about more direct applications as opposed to something like that.)

REPLY [6 votes]: With 562 citations on Mathscinet, it's hard to summarize all of the applications of this influential paper! One important one was the extension of the Atiyah-Bott results to the setting of parabolic bundles and also to bundles over 2-dimensional orbifolds. The latter, coupled with calculations by Fintushel and Stern, permits one to calculate the Instanton Floer homology of Brieskorn homology spheres. These are Seifert-fibered spaces with 3 or more fibers; once you get up to 5 fibers you need to calculate the homology of the space of SU(2) representations of some 2-dimensional orbifolds. 
I wouldn't underestimate the broader influence as well; the use of equivariant Morse theory for one, and also the many ideas that were used in Donaldson's work on 4-manifold invariants.