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.)
Best Answer
They observed that the algebraic concept of stability is equivalent with the analytic concept of Yang-Mills connection. This has a variational characterization opening the door for the usage of Morse theory. In particular they used topological methods to solve an algebraic0geometric problem.
A bit later, Donaldson proved a similar result stating that on algebraic surfaces the concept of stable bundle is equivalent with the concept of instanton. Then he used algebraic geometric methods to solve a topological problem, the computation of Donaldson invariant for certain algebraic surfaces.
If you open the book of Donaldson on instantons and 4-manifold you will see how heavily he was influenced by the set-up in Atiyah and Bott paper.
So the contribution of Atiyah-Bott paper is twofold: they solve an algebraic geometric problem and they introduced this new point of view that turned out to be extremely fertile.