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Best books for mathematical background?

I want to learn contemporary mathematical physics, so that, for example, I can read Witten's latest paper without checking other sources again and again to find some basic definitions and theorems. I know it need a long time and intensive efforts, but are there any good books related so that I can follow them in one or two years? I have learned physics theories that come before the quantum field theory, including general relativity. And I know differential geometry, category, etc.

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marked as duplicate by David Z Jun 26 '12 at 19:34

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

Mathematical physics is just too broad at this point. If you pick a particular area, people can point you to useful references. As a start, you can't go wrong reading Nakahara's, "Geometry, Topology and Physics, and Nash's "Differential Topology and Quantum Field Theory". And learn quantum field theory. – Aaron Dec 15 '11 at 1:05
I don't think you can expect useful advice without narrowing things down, you'll just get everybody's favorite math book, which will send you on a wild goose chase. There are many math physics books because each one had different purpose, you'd have to decide what is yours. – user566 Dec 15 '11 at 2:08
This question seems rather vague and not well suited to this stack exchange. – Benjamin Horowitz Dec 16 '11 at 16:03

3 Answers 3

Start by reading Witten's early papers. He wasn't able to expect his audience to know all the math already, so he often did a very nice job of explaining it.

For example,

  • Supersymmetry Algebras That Include Topological Charges
  • Search for a Realistic Kaluza-Klein Theory
  • A Simple Proof of the Positive Energy Theorem
  • Constraints on Supersymmetry Breaking
  • Dynamical Breaking of Supersymmetry
  • Global Aspects of Current Algebra
  • Current Algebra, Baryons, and Quark Confinement
  • Non-Abelian Bosonization in 2d
  • Strings on Orbifolds
  • Supersymmetry & Morse Theory
  • Baryons in the 1/N expansion
  • Verlinde Algebra & the Cohomology of the Grassmannian
  • Gravitational Anomalies
  • ...
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Great! what papers would you recommend to start with? – Larry Harson Aug 12 '13 at 23:18
@LarryHarson I've listed a few of Witten's papers in the main body of the text. I don't think I can sensibly recommend starting with any particular paper, certainly not without knowing what you're interested in studying. – user1504 Aug 13 '13 at 14:34

I have not read Witten's papers, but if you're willing to look at a senior undergrad, first year grad, text, I'd suggest Hassani's. It is very broad in scope, and provides a good introduction to a number of areas of mathematical physics.

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This is quite late. I found Mathematical Perspectives on Theoretical Physics: A Journey from Black Holes to Superstrings quite suitable for what you ask.

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The Table of Contents of this book does look very tempting. How did you perceive this book? The Amazon reviews seem to be very negative – Michael Mar 5 '12 at 11:20
I do not work in String theory. So I do not wish to say anything about the review in Amazon. I use the book as a reference and handbook for the mathematics. – Vijay Murthy Mar 6 '12 at 10:11