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I've been studying 3 and 4 manifold topology and it seems to me that lots of very powerful invariants come from a mysterious place called "gauge theory". When I peer into this place, I am confronted with a lot of language which doesn't seem native to the language I've been used to (I've read a book on differential topology, various papers on heegaard splittings, Hatcher's notes on 3-manifolds, and the Kirby Calculus part of Gompf+Stipsicz).

Is there a good introduction to gauge theory which assumes (something close to) my basic topological knowledge? I would anticipate that someone might have tried to tackle building this kind of scaffold.

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  • $\begingroup$ Something like this might fit the bill: pdmi.ras.ru/EIMI/2013/Cohomology/lect/mm.pdf $\endgroup$ Feb 19, 2019 at 1:05
  • $\begingroup$ @CheerfulParsnip A lot of that looks uncomfortably like physics to me! The first 20 pages are probably a good first pass at understanding the ideas (but I can never read the index notation). $\endgroup$
    – user98602
    Feb 19, 2019 at 1:21
  • $\begingroup$ @MikeMiller it's definitely a tough climb. $\endgroup$ Feb 19, 2019 at 1:28
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    $\begingroup$ Now that I think of it, I remember liking Morgan's "Introduction to Gauge Theory", in "Gauge Theory and the Topology of Four-Manifolds". I don't remember that it gave much attention to making you comfortable with the analysis. $\endgroup$
    – user98602
    Feb 19, 2019 at 1:40

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I do not think there is a quick route to understanding here, though there might be some notes that give a rough idea (as in those suggested in the comments). I am not familiar with any brief notes of that type, so I will not talk about them.

To understand Donaldson's (instanton) invariants, the Seiberg-Witten invariants, or either of their 3-dimensional Floer homology counterparts, you need to know some differential geometry of connections (and in the case of SW invariants, some spin geometry) and to understand the rigorous construction, you need to know some analysis (Sobolev spaces and some juggling with Sobolev multiplication theorems) that's a little more complicated than the kind of linear PDE involved in Hodge theory. Many sources on gauge theory cover these, but often very briefly, and expect you to be comfortable already or get comfortable as you go.

I think the easiest thing to do here is for me to simply say what my favorite sources are on each of these topics, and give them a sort of difficulty level. In almost all of these, the discussion of Sobolev spaces is very brief, providing no more than what is needed for the definition of the invariants.

4D

Instantons:

Freed and Uhlenbeck's "Instantons and 4-manifolds". This book is primarily concerned with proving Donaldson's diagonalizability theorem (and discussing Taubes' extension to 4-manifolds with periodic ends). It is rather gentle.

Donaldson and Kronheimer, "The Geometry of 4-Manifolds". This covers the definition of Donaldson's invariants for closed 4-manifolds (as well as the diagonalizability theorem), and some calculations for complex surfaces. It is very detailed but also requires quite a bit of effort from the reader; I think it was about two years before I worked my way through the whole thing. (Of course, this wasn't all I was thinking about for that time.)

Seiberg-Witten equations (whose solutions are sometimes called "monopoles"):

Moore: "Lecture Notes on Seiberg-Witten Invariants". This starts from scratch and teaches you the relevant spin geometry. The analysis of the moduli spaces in the Seiberg-Witten setting is less intensive than in the instanton setting, and so this gets to the heart of the matter more quickly. This covers a proof of Donaldson's diagonalizability theorem, the definition of SW invariants, and some calculations.

Morgan: "The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds". More or less the same goals as the above. I remember that I slightly prefer this book, but to be honest, I don't remember why. It would be handy to have them side-by-side.

Salamon: "Spin Geometry and Seiberg-Witten Invariants", available here (warning: 600 page PDF!)

It includes discussions of calculations, including connected sum theorems, symplectic manifolds, and various applications to topology and geometry. The level of detail is exceptional, including appendices on Fredholm operators, Sobolev spaces, and elliptic regularity theorems, and an extended discussion of all the relevant spin geometry. This was probably the first thing I read that I really understood gauge theory from; the level of detail made it a rather leisurely read for me.

3D

Instantons:

Donaldson, "Floer homology groups in Yang-Mills theory". Discusses the definition of instanton Floer homology for homology 3-spheres, as well as some discussion of 3-manifolds equipped with "non-trivial admissible bundles".

Floer's original paper, "An instanton invariant for 3-manifolds", is very readable (IMO), especially if you know some of the 4-dimensional story already. I read this long before finishing Donaldson's book.

Seiberg-Witten:

There is one main option here, and it is a gorgeous one: Kronheimer and Mrowka's book "Monopoles and 3-manifolds".

It is self-contained, and teaches you all of the necessary spin geometry; the first few chapters start by telling you the story before getting into the nitty-gritty. A lot of it is written in sufficient generality to be usable for most situations in gauge theory, and I have carried a copy with me for the last 3-4 years. A lot is covered, in a lot of detail.

Kim Frøyshov has a book on an alternative definition of monopole homology for rational homology spheres. I unfortunately haven't read it yet, but have meant to for some time. I am familiar with his construction (but not all the technical details), and by its nature it will be somewhat less technical than Kronheimer and Mrowka's.


For almost all of these, I would like to give a piece of advice: if something is hard to understand, move on until later. It will eventually become clear whether or not you really need to understand the argument you were looking at. If you do, you can go back, feeling more comfortable with the material in general. This is especially true if you're mostly interested in learning the story.

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    $\begingroup$ I guess an answer to the title question is: Salamon's book is a great place to start. If you don't care about the analysis and just want to understand the idea, grab any of these books - in the precise area you care about - and read through the introductory chapters about what they prove. Ultimately, the goal is to write down some "moduli spaces" and extract numbers from them, and the analysis is to show that these can be made to be smooth manifolds (more or less), and then to understand their compactness properties. $\endgroup$
    – user98602
    Feb 19, 2019 at 1:36
  • $\begingroup$ Thank you very much for the details...I am gonna bookmark this for future references :) $\endgroup$ Feb 20, 2019 at 1:51

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