16

Please do not ignore the other author, Peter Kronheimer. Based on all of the material I've read, I do not agree with your belief about the book. I think it is more detailed than you will find elsewhere which covers all of that material. Here are some useful alternatives, though:
Take that book and replace the structure group $SU(2)$ by $SO(3)$. This route ...

11

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 ...

answered Apr 14 '18 at 8:50

Liviu Nicolaescu

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9

It's been a while but I remember that I found John Morgan's "An introduction to gauge theory" quite helpful when I was first trying to read about 4-manifolds and gauge theory. While it doesn't go nearly as far as Donaldson-Kronheimer, it does give a gentle introduction to gauge theory. Here's the complete reference:
Morgan, John W., An introduction to gauge ...

6

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 ...

6

We discussed some conjectural implications in Section 4.2 of the paper. I wouldn't say that the category of sheaves with nilpotent singular support was necessarily the "right" category to consider from a gauge theoretic point of view. Instead I'd say that if one considers the equivalence for the whole category of coherent objects (or its completion) the ...

5

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 ...

4

$\newcommand{\A}{\mathscr{A}}$ $\newcommand{\G}{\mathscr{G}}$ Denote by $\A$ the space of connections, by $\A_-$ the space of ASD connections and by $\G$ the gauge group. For ssimplicity I will not keep track of various Sobolev decorations. The moduli space $\newcommand{\M}{\mathscr{M}}$ $\M$ is defined as a set by the equality
$$\M=\A_-/\G\cdot A. $...

answered Nov 19 '17 at 12:54

Liviu Nicolaescu

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4

A quite "low-brow" book you might find enjoyable reading is The Wild World of Four manifolds, which surveyed the whole theory beginning from handle body decomposition. It has a long list of reference books you can use for further reading. The material may be out of date by now, though. If you can provide more detailed feedback on where you got stuck, I am ...

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