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I have finished my qualifying exam for the PhD in Math, and would like to survey through different advanced Mathematics topics, particularly focusing on Representation Theory and Mathematical Physics.

I would love to ask if someone can recommend me TWO books on the following topics, in particular,

1) the first book is the "Bible" that everyone working in the field should know, and

2) the second book is a "good" introductory book that hopefully provides many examples to demonstate the theory, (since in my opinion most classical "bible" is too abstract and concise to spend time explaining thoroughly the concept...)

-Langlands Program / Duality

-Affine Lie Algebra / Kac Moody / Loop Groups ...

-Vertex (Operator) Algebra

-Non-compact Quantum Groups / Hopf Algebra

-Non-commutative Geometry

-Conformal Field Theory / Quantum Field Theory

-Absolute Galois Group

-Cohomology Theory (in Algebraic Geometry)

-Hyperbolic Geometry

-All the physics involving those SO(n,m), SL(n,C), SU(n) in different Spacetime, etc etc (I don't know how to phrase this topic...Gauge Theory?)

Really appreciate your help.

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4 
 This question is too unfocused, I think. – Qiaochu Yuan Feb 9 2011 at 22:49
2 
 Please see the FAQ on how to ask a good question. In any case, if asked well, and in a focused way, this should be a community wiki question. – David Roberts Feb 9 2011 at 22:58
there are references above to "the theory". The theory of what? Representation theory is such a vast area that I think you need to make your list more focused if you want to get useful answers here. – Yemon Choi Feb 9 2011 at 23:16
For instance, it seems like what you really want is a book on "Representation Theory for Theoretical Physicists"; if that is the case, then a question concentrating on that is more likely to receive helpful answers, than the way you've currently asked it. – Yemon Choi Feb 9 2011 at 23:18

closed as not a real question by Andres Caicedo, David Hansen, Qiaochu Yuan, David Roberts, Tony Huynh Feb 9 2011 at 23:07

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