Book covering Topology required for physics and applications

Question

I am a physics undergrad, and interested to learn Topology so far as it has use in Physics. Currently I am trying to study Topological solitons but bogged down by some topological concepts. I am not that interested for studying it for its own sake. Please could you mention the topics of Topology that are required in Physics? Could anyone recommend me a book that deals with these topics and also some applications to Physics. I have taken an introductory course in Real Analysis (Sherbert, Apostol, etc), and have no knowledge of complex analysis.

This post imported from StackExchange Physics at 2014-04-05 16:34 (UCT), posted by SE-user ramanujan_dirac

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Related: Book Recommendations for Topology    

Answer 1

If you want to learn topology wholesale, I would recommend Munkres' book, "Topology", which goes quite far in terms of introductory material.

However, in terms of what might be useful for physics I would recommend either:

• Nakahara's "Geometry, Topology and Physics"
• Naber's "Topology, Geometry and Gauge Fields: Foundations"

Personally, I haven't read much of Nakahara, but I've heard good things about it, although it may presuppose too many concepts. I've read selections of Naber and it seems fairly well written and understandable and starts from first principles, but again, it may not focus as much on the fundamentals, if that's what you're looking for.

This post imported from StackExchange Physics at 2014-04-05 16:34 (UCT), posted by SE-user Nilay Kumar

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What about Topology for physicists by Schwarz?

This post imported from StackExchange Physics at 2014-04-05 16:34 (UCT), posted by SE-user ramanujan_dirac

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Sorry, I haven't heard anything of that book. However, judging by the absolutely ridiculous price on Amazon... Anyway, I was also flipping through Nash and Sen's book, and it seemed to treat topology in a very intuitive and clear manner, although at a mathematical price - Amazon reviewers claim that it isn't too mathematically rigorous/comprehensive.

This post imported from StackExchange Physics at 2014-04-05 16:34 (UCT), posted by SE-user Nilay Kumar

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I enjoyed reading Nash and Sen, It suited my taste, being less formal, and more intuitive. Nakahara is nice. Schwarz seemed good at first glance, but I havent read it.

This post imported from StackExchange Physics at 2014-04-05 16:34 (UCT), posted by SE-user Prathyush

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You also asked about topics in topology relevant for physics. Apart from the basic definitions and so on, one of the most applied concepts is Homotopy. It is beautiful in itself, and it formalizes the concept of winding numbers to higher dimension. In physics it is commonly used to enumerate the topological solitons present in your theory.There are others, but I found Homotopy to be very important and useful.

This post imported from StackExchange Physics at 2014-04-05 16:34 (UCT), posted by SE-user Prathyush

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I enjoyed Naber's first gauge theory book. Of course, there are many excellent texts to read on topology and geometry. One which might be a good thing to have sitting around to come back to in years to come is

http://books.google.com/books/about/Anomalies_in_quantum_field_theory.html?id=567vAAAAMAAJ

Anomalies in Quantum Field Theory by Reinhold A. Bertlmann.

There are about 100 pages of math at the outset of this text. It gives you a nice birds-eye view of math. Naturally, you will need deeper math books to really "get" the math...

Answer 2

You can try to read the notes of David Tong (available on-line so just Google it) on solitons and instantons but surely some basic topology is needed. One need's to have a heuristic understanding of e.g. Hopf fibration, homotopy group, equivalence classes to understand solitonic solutions. At least from my experience.

Answer 3

Don't hurry ramanujan, learn basic mathematical methods first (from Sadri-Hassani's "Mathematical Physics" for instance). Then the standard reference for you to learn grad-level mathematics would be Nakahara's "Geometry, Topology and Physics". If you think it's too much, you're right; this is a very serious advanced topic. But if you want to quickly pick some basic ideas, check out the 10th chapter of Ryder's "Quantum Field Theory". An advanced and physically oriented discussion would be found in Coleman's "Aspects of Symmetry".

This post imported from StackExchange Physics at 2014-04-05 16:34 (UCT), posted by SE-user Kachal

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What about Topology for physicists by Schwarz?

This post imported from StackExchange Physics at 2014-04-05 16:34 (UCT), posted by SE-user ramanujan_dirac