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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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    $\begingroup$ 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. $\endgroup$
    – Aaron
    Dec 15, 2011 at 1:05
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    $\begingroup$ 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. $\endgroup$
    – user566
    Dec 15, 2011 at 2:08
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    $\begingroup$ This question seems rather vague and not well suited to this stack exchange. $\endgroup$ Dec 16, 2011 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
  • ...
  • $\begingroup$ Great! what papers would you recommend to start with? $\endgroup$ Aug 12, 2013 at 23:18
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    $\begingroup$ @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. $\endgroup$
    – user1504
    Aug 13, 2013 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.


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

  • $\begingroup$ 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 $\endgroup$
    – Michael
    Mar 5, 2012 at 11:20
  • $\begingroup$ 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. $\endgroup$ Mar 6, 2012 at 10:11