# A gentle introduction to CFT

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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 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.)

This post imported from StackExchange Physics at 2014-03-06 11:12 (UCT), posted by SE-user Federico Carta

recategorized Apr 24, 2014
I've seen that tend to ask too many questions in one. In general, you should avoid this practice so people can answer one at a time. And wrt. the first question, doesn't Wikipedia answer it?

This post imported from StackExchange Physics at 2014-03-06 11:12 (UCT), posted by SE-user jinawee
Thanks for the note. From now on I will ask only one question at a time. Do you think I can leave this as it is, or should I edit and split it?

This post imported from StackExchange Physics at 2014-03-06 11:12 (UCT), posted by SE-user Federico Carta
Wait until a moderator sees it.

This post imported from StackExchange Physics at 2014-03-06 11:12 (UCT), posted by SE-user jinawee
@Qmechanic, would it work if I split this post in 5 different questions?

This post imported from StackExchange Physics at 2014-03-06 11:12 (UCT), posted by SE-user Federico Carta
In general Yes, jinawee is right: only one question per post; but unfortunately in this case it seems to be No. All five subquestions seem closable for various reasons (such as, e.g., question answered by a simple wiki-search, primarily opinion-based question, list-type question).

This post imported from StackExchange Physics at 2014-03-06 11:12 (UCT), posted by SE-user Qmechanic
Well I know... I asked it there after this topic was closed. Thanks anyways for pointing it out.

This post imported from StackExchange Physics at 2014-03-06 11:12 (UCT), posted by SE-user Federico Carta
The subquestions specify what the OP is looking for further, which does not mean that the question is too broad or something...

This post imported from StackExchange Physics at 2014-03-06 11:12 (UCT), posted by SE-user Dilaton

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Which is the definition of a conformal field theory?

"The" definition does not exist. There are many (related but not completely equivalent) definitions, depending on the perspective. Some of them are mentioned in the review here.

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 instantons etc?)

Instantons cannot arise in CFT. Supersymmetry can, but much of CFT can be understood without it. A knowledge of the meaning of free quantum fields and the associated machinery is essential, and understanding the Wightman axioms is helpful.

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?)

You need a good acquaintance with complex analysis (Laurent series) and with semisimple Lie algebras and their representations. If you are only interested in CFT on the torus, neither Riemann surfaces nor algebraic geometry is needed. if you are interested in CFT at genus $>0$ (needed, e.g., for string theory) you need some basics in both areas, and on category theory.

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

For the case of 2 dimensions, I recommend the review article by Fuchs, the lecture notes by Ginsparg, and the books by Kac (Vertex algebras for beginners) and di Francesco (Conformal field theory). In higher dimensions I recommend Rychkov's  Lecture notes on CFT.

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

At a critical point, most thermodynamic systems are described by a corresponding CFT.  See, e.g., Lecture 5 in the book by Ginsparg mentioned above. The dynamics of the quantum Hall effect is also governed by CFT.

answered Apr 29, 2015 by (14,537 points)

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