# A gentle Introduction on CFT

1. Oct 28, 2013

### Fedecart

Hello everybody! Wanted to ask a couple of questions...

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.)

2. Oct 28, 2013

### DimReg

Presumably someone with more knowledge of the subject will come by, but for a quick intro to CFT almost any string theory textbook has a chapter on it (if I recall correctly, most of the interest in studying CFTs comes from their connection to string theory). It's probably not worth purchasing a textbook just for that one chapter, but if you borrow a copy or get it from a library, that's a place you can start.

3. Oct 28, 2013

### fzero

A CFT is a field theory that is invariant under the group of conformal transformations on spacetime. This is the group of transformations that leaves the metric invariant up to a scale factor:

$$g'_{\mu\nu}(x') = \Omega(x) g_{\mu\nu}(x).$$

It's best to look at one of the references below for more discussion.

To start, there aren't too many prerequisites beyond basic quantum field theory. You would want to be familiar with Poincare invariant QFT, group representations, correlation functions of local fields and the renormalization group. It is not strictly necessary to know about supersymmetry or nonperturbative QFT, though a familiarity with advanced QFT would be helpful for studying more advanced CFT topics.

It is necessary to be very familiar with the Laurent series, the residue theorem and modular transformations. It can be helpful to be familiar with differential topology of the torus, but it may be possible to follow the discussion in a CFT reference without having a deep background in topology and geometry. Again, for more advanced topics, more math would be helpful. It's probably best to just dive in and figure out what you get stuck on, referring to a book like Nakahara as needed.

A standard free reference is Ginsparg's lectures. There is a list of additional online articles here.

CFT is relevant to the discussion of scale-invariant quantum systems, so is deeply related to discussions of the renormalization group, which is also a feature of statistical mechanics of critical phenomena. Most of the textbooks, like Di Francesco et al (the Big Yellow Book) will have a more complete discussion of applications, but there are some obvious examples like the quantum Hall effect from condensed matter.

4. Apr 29, 2015