22 votes
Accepted

About Simon Donaldson's book on four dimensional manifold

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 ...
Chris Gerig's user avatar
  • 17.1k
16 votes
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A non-Abelian de Rham complex?

What is being described in the main post is simply the de Rham (crossed) complex valued in a Lie group (not necessarily commutative). See, for example, Section 6.2 in Anders Kock's Synthetic Geometry ...
Dmitri Pavlov's user avatar
15 votes

What are the applications of the Atiyah-Bott Yang Mills paper?

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 ...
Liviu Nicolaescu's user avatar
11 votes

About Simon Donaldson's book on four dimensional manifold

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 ...
Stefan Behrens's user avatar
8 votes

A non-Abelian de Rham complex?

Unfortunately, things are subtle and terrible (at least, compared to the abelian case). I wrote a bit about this in an unpublished article. One of the surprising, terrible features is that you can ...
Matt Noonan's user avatar
  • 3,984
8 votes

What are the applications of the Atiyah-Bott Yang Mills paper?

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 ...
ThiKu's user avatar
  • 10.3k
7 votes

Yang-Mills over surfaces

In general it is not in the center of $\frak g$. The curvature of a Yang-Mills connection in this dimension is parallel which means it commutes with the image of holonomy representation which need ...
Tom Mrowka's user avatar
  • 3,004
6 votes
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Implications of gauge symmetry breaking on the spectral side of geometric Langlands?

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 ...
Chris Elliott's user avatar
6 votes

What are the applications of the Atiyah-Bott Yang Mills paper?

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 ...
Danny Ruberman's user avatar
6 votes

About Simon Donaldson's book on four dimensional manifold

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 ...
Bombyx mori's user avatar
  • 6,141
5 votes
Accepted

Deformation-Obstruction Theory of YM Instantons

$\newcommand{\A}{\mathscr{A}}$ $\newcommand{\G}{\mathscr{G}}$ 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 ...
Liviu Nicolaescu's user avatar

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