Prerequisites for ‘Quantum field theory and representation theory: a sketch’ [arXiv:hep-th/0206135]

I'm interested in reading Dr. Peter Woit's article, Quantum field theory and representation theory: a sketch [hep-th/0206135].

What math and physics background would be needed?

(A list of topics from the two fields would be more than sufficient.)

With great appreciation and regards.

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Why two downvotes? The question seems reasonable to me. –  Pete L. Clark Aug 6 '12 at 18:06
Sadiq, it'd help if you could give us a sense of your mathematical experience until now, since the list of prerequisites for quantum field theory ideally amounts to an undergraduate degree in physics, and for representation theory, not much less than one in math. –  Kevin Carlson Aug 6 '12 at 18:18
(Formal) Math: Up to several-variable calculus Physics: First-year physics (Informal) Math: Very basic and sketchy knowledge of real & complex analysis, some differential geometry. Physics: E&M, first-semester QM. –  Sadiq Ahmed Aug 6 '12 at 18:19

Most of the math prerequisites are then more along the line of differential topology. I won't suggest a specific text, but you need to know about vector bundles and fiber bundles, and eventually probably about general sheaves over a space. From that last you'll be able to define the group $K_0$ which is the initiation of K-theory, a central topic of the paper you're interested in. You'll also want some general representation theory, for which I can recommend the book of Fulton and Harris. They may also have all the background material specifically on Lie algebras and Lie groups you need. The last large mathematical area that would come in handy is some category theory, since K-theory is going to be developed in terms of functors-for that you can get Steve Awodey's book Category Theory for free from his website.