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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.)

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    $\begingroup$ I'm not enough of an expert to answer, so I'll leave a comment that others can correct. I think the background is that by 1983 algebraic geometers had a lot of success solving problems using moduli spaces of sheaves, and Atiyah figured these techniques ought to have topological / analytic counterparts. So far as I am aware this paper solves the first big moduli problem in differential geometry, and the fact that it uses a crucial ingredient from physics is just icing on the cake. So the paper is a "proof of concept" which paved the way for further developments in low dimensional topology. $\endgroup$ – Paul Siegel Apr 14 '18 at 1:10
  • $\begingroup$ Not sure about what you mean by "just icing on the cake" and maybe you meant that this was a proof of concept that Yang-Mills theory could be used to study moduli spaces? My recollection is that this was done not long after mathematicians, led by Simons and Singer, first realized that Yang-Mills theory was really about connections on principal bundles. Atiyah and Bott, I believe, decided that the easiest way to learn the stuff was to work it all out on surfaces, rather than 4-manifolds. $\endgroup$ – Deane Yang Apr 14 '18 at 3:31
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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.

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Hitchin 1987 extended the work of Atiyah-Bott to study the topology of the moduli space of Higgs bundles on $\Sigma$ via the Yang-Mills functional. Simpson 1988 proved that this moduli space agrees with the character variety of representations from $\pi_1\Sigma$ to $SL(n,{\mathbb C})$. Lateron this approach was generalized to study the moduli space of representations from $\pi_1\Sigma$ to a reductive Lie group.

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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.

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