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 ...
16
votes
Accepted
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 ...
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 ...
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 ...
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 I wrote a bit about this in an unpublished article.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 ...
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 ...
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 ...
6
votes
Accepted
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 ...
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 ...
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 ...
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 ...
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