Good textbook or lecture notes on Seiberg-Witten theory.

Question

I am looking for a good introductory book for Seiberg-Witten theory. The only textbook I have now is Morgan's "The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds". This book is self-contained and concise and definitely a good book, but I am lost before section 4 "Seiberg-Witten moduli spaces" due to my poor background.

I would appreciate it if you could introduce me a good textbook or introduction paper for this subject. I am also happy if anyone knows a good reference for spin bundles and Clifford bundles. I am aware of the book "Spin Geometry" by the way.

Answer 1

My advisor Michael Hutchings and his advisor Cliff Taubes wrote a brief note on this which is where I definitely suggest starting: http://math.berkeley.edu/~hutching/pub/tn.pdf
If you're getting stuck around the "moduli space" part, then I suggest taking a step back and understanding the general picture of counting pseudoholomorphic curves, from McDuff & Salamon's classic book on Symplectic Topology.

If this was enough overview, then the next step (and perhaps the last that you'd need) is Salamon's Spin Geometry and Seiberg-Witten Invariants, which deals with all the required background plus the thorough development of the theory, packed with a ton of useful/friendly info.

Answer 2

Perhaps these references will help you find what you are looking for.

Books:

Seiberg-Witten Theory and Integrable Systems Andrei Marshakov

Lectures on Seiberg-Witten Invariants (Lecture Notes in Mathematics) John D. Moore

The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds (Mathematical Notes, Vol. 44) John W. Morgan

Notes on Seiberg-Witten Theory (Graduate Studies in Mathematics, Vol. 28) Liviu I. Nicolaescu

Free Notes:

staff.ustc.edu.cn/~craigvan/SW-theory11.pdf

https://www.nd.edu/~lnicolae/swnotes.pdf

www.people.fas.harvard.edu/~xiyin/SW.pdf

math.berkeley.edu/~hutching/pub/tn.pdf

matwbn.icm.edu.pl/ksiazki/bcp/bcp39/bcp3925.pdf

www.math.ucsb.edu/~moore/seibergwittenrev2edition.pdf

www.het.brown.edu/~nastase/sw.pdf

matematicas.uniandes.edu.co/summer2001/1999/l7.ps

www.mathematik.uni-tuebingen.de/~thilo/marco.ps

Answer 3

The material in chapters 1-3 of Morgan's book are really just prerequisite knowledge for anyone even wanting to start learning Seiberg-Witten theory (the actual beginning of the coverage of Seiberg-Witten theory is chapter 4). Morgan's coverage of Clifford algebras, spin geometry, and Dirac operators is very quick and does not go into much detail. Therefore if you have not had much exposure to this material before, it is very understandable if you get lost before Morgan even tells you what the Seiberg-Witten equations are.

Because of this, I would advise you to look at a text that spends a fair amount of time developing spin geometry. You have mentioned you are aware of Lawson and Michelson's Spin Geometry. This is certainly the best book on spin geometry that I know of, and it is what I personally used to supplement Morgan's book when I started learning Seiberg-Witten theory, but perhaps there are too many details in Lawson and Michelson that you won't need at this point. For less detail than Lawson and Michelson, you might consider looking into these lecture notes on spin geometry (lectures 7 and 8 won't be necessary for Seiberg-Witten theory).

As for other texts on Seiberg-Witten theory that might be easier for you, if you can get your hands on Salamon's Spin Geometry and Seiberg-Witten Invariants, I think it would be very good for you as he takes the time to develop the prerequisite knowledge that Morgan blazes through. Perhaps the easiest book on Seiberg-Witten theory is Moore's Lectures on Seiberg-Witten Invariants, but I didn't really enjoy this book as everything seemed too watered down and the prerequisite material on Clifford algebra and spin geometry is only developed for $4$-manifolds, leaving you with little idea of where it actually comes from in general.

On a different note, I think you should avoid Nicolaescu's Notes on Seiberg-Witten Theory. It is the most comprehensive book available for Seiberg-Witten theory on $4$-manifolds, but I think all the details and complicated analysis would overwhelm anyone who is just learning the material for the first time. I would also avoid Marcolli's Seiberg-Witten Gauge Theory, as it covers the prerequisite material you are having a hard time with even faster than Morgan does.