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1) Which is the definition of a conformal field theory?

2) Which are the physical prerequisites one would need to start studying conformal field theories? (i.e Does one need to know supersymmetry? Does one need non-perturbative effects such as solitons, instantons etc?)

3) Which are the mathematical prerequisites one would need to start studying conformal field theories? (i.e how much complex analysis should one know? Does one need the theory of Riemann Surfaces? Does one need algebraic topology or algebraic geometry? And how much?)

4) Which are the best/most common books, or review articles, for a gentle introduction on the topic, at second/third year graduate level?

5) Do CFT models have an application in real world (already experimentally tested) physics? (Also outside the high energy framework, maybe in condensed matter, etc.)

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closed as too broad by Carlo Beenakker, Ricardo Andrade, Qiaochu Yuan, Andres Caicedo, Kevin Walker Oct 28 '13 at 23:44

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs.If this question can be reworded to fit the rules in the help center, please edit the question.

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@Federico. You wrongly assume that there is a unique mathematical definition of CFT. There are many (I know three). Probably all good. And people don't know how to compare them (at a mathematical level of rigor). The mathematical prerequisites are very different depending on which approach to CFT you decide to study. Also, there is an important distinction between "chiral CFT" and "full CFT": those are two completely different things (but not unrelated). –  André Henriques Oct 28 '13 at 23:40
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This is probably irrelevant since this question is already closed, but I'll just note that it's a cross-post of a question which was put on hold on Physics SE. –  Logan Maingi Oct 29 '13 at 1:04
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@Logan Maingi I believed that different places have different rules on what can or can not be posted. Is it really bad to crosspost if it did not work there? Please I am not trying to be arrogant, just trying to understand, since it is my first time in both of these places –  Federico Carta Oct 29 '13 at 1:47
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What's with the hate for this question? Regarding (1), there's a paper of Segal titled ``The definition of conformal field theory'' that might be relevant. As for (2), (3), (4), I would very much like an answer myself. I also think (2), (3), (4) are just reformulations / aspects of / clarifications of the same underlying question, so the complaint that there's too many questions here is just absurd. –  Vivek Shende Oct 29 '13 at 5:06
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@Federico: For references try Gaberdiel's review paper on conformal field theories. For commented pointers to the literature see here ncatlab.org/nlab/show/conformal+field+theory Or for one rigorous definition and derivation of the full theory from the first principles see the book by P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory (Springer, 1997) (and this is my source). –  Irina Oct 29 '13 at 10:38