Mathematics Ph.D Student trying to learn Conformal Field Theory

[Tom Gilroy]

Hi everybody,

I'm not entirely sure if this should be posted here or in the Quantum Physics section, if a moderator feels it would be more suitable there, please feel free to move it.

As the title indicates, I'm a mathematics Ph.D student (studying Vertex Operator Algebras) and I'm interested in learning conformal field theory. This thread is essentially asking for advice on how to do this.

I do have some background in physics, and I imagine it would be helpful if I gave some details.

I studied physics as an undergraduate for 2 years, my 2nd year would have been at the level of Halliday, Resnick and Krane. Later I took a two semester couse in quantum mechanics and a one semester course in general relativity (the QM course seems to have been at a slightly higher level than most undergraduate courses, while the GR course seems to have been at the "average" level). I'm comfortable with classical mechanics, however I've never spent much time on electromagnetism.

I achieved very good results in all courses.

My background in mathematics is considerably stronger.

I would assume that an obvious prerequisite to CFT would be QFT, but I'm not sure what level I should study QFT to, or what topics are relevant. Would anybody have any textbook recommendations for QFT? Are there any other prerequisites I should be aware of? Statistical mechanics maybe? Or should I just start with CFT straight away, and learn QFT as I come to need it? Am I right in assuming that Di Francesco, Mathieu and Senechal is the book to learn CFT from? Or would anybody have any other recommendations?

 

[negru]

You can also try http://arxiv.org/abs/hep-th/9108028 for an intro, it's more compact than THE BOOK.

How much qft you need to know depends on how much cft you want to know. The bare minimum is obviously learning what correlation functions are.

For qft I'd recommend Srednicki (free at http://www.physics.ucsb.edu/~mark/qft.html) . Your background (qm and gr) should be enough for that. I think just the first chapter on scalar field theory should give you most of the tools you need to begin with.

 

[negru]

(btw you can probably skip chapter 3 on stat mech, unless you're particularly interested in that)

 

[Tom Gilroy]

Thank you for the response.

I'll begin working through the first chapter of Srednicki soon. Thank you for the link to Ginsparg's lectures also. They certainly look a lot more digestible than the book!

 

[haushofer]

I think by far the best book to learn CFT is indeed still DiFrancesco et. al. It's very big, but that's because it's very detailed; everything you want to know is in it including the QFT. But it would be wise to study some QFT apart from that; Srednicki is indeed a nice book which immediately gets you involved into renormalization. Personally I also like Peskin&Schroeder.

 

[martinbn]

For QFT you may want to take a look at Folland's "Quantum Field Theory. A Tourist Guide for Mathematicians". Written by a mathematician for mathematicians.

 

[Kevin_Axion]

If you wait a a few months there will be a lecture series on specifically this topic in the Perimeter Scholars International lecture archive: http://pirsa.org/C10017. The curriculum is as follows:http://www.perimeterscholars.org/course-curriculum.html.

 

[Tom Gilroy]

I'd first like to apologise for breaking the rules by posting in the wrong forum, however, I don't intuitively see how this question is more appropriately discussed under the banner of "academic guidance." I did also read read the forum guidelines and the rules regarding the "Beyond the Standard Model" forum, neither of which explicitly said that a thread of this kind is inappropriate in that forum. Regardless, I will do my best to ensure it doesn't happen again.

Thank you all for your input, it has been very helpful.

Folland's book seems like it might be useful, though it seems like it might be written from an analysis based standpoint. That wouldn't be an issue, but I'm more interested in the algebraic aspects. I will definitely look further into it.

Considering the availability of the PSI lectures in a few months, would you all advise learning QFT from Srednicki and the PSI video lectures for the moment, then moving onto learning CFT from Francesco et al., Ginsparg's notes and the PSI video lectures when they become available?

 

[crackjack]

Gaberdiel's intro to CFT (http://arxiv.org/abs/hep-th/9910156v2) is a good accompaniment to have on the side.

 

[Tom Gilroy]

Thank you!

 

[xepma]

I'd first like to apologise for breaking the rules by posting in the wrong forum, however, I don't intuitively see how this question is more appropriately discussed under the banner of "academic guidance." I did also read read the forum guidelines and the rules regarding the "Beyond the Standard Model" forum, neither of which explicitly said that a thread of this kind is inappropriate in that forum. Regardless, I will do my best to ensure it doesn't happen again.

Thank you all for your input, it has been very helpful.

Folland's book seems like it might be useful, though it seems like it might be written from an analysis based standpoint. That wouldn't be an issue, but I'm more interested in the algebraic aspects. I will definitely look further into it.

Considering the availability of the PSI lectures in a few months, would you all advise learning QFT from Srednicki and the PSI video lectures for the moment, then moving onto learning CFT from Francesco et al., Ginsparg's notes and the PSI video lectures when they become available?

Do not go through the entire book of Srednicki if your primary goal is CFT. The reason is that QFT in the conventional way has some aspects which are fairly absent in CFT. For instance:

-perturbation theory
-renormalization
-running coupling constants
-gauge theory
-symmetry breaking

are usually large topics that need to be tackled in a course on QFT. But for a first treatment on CFT these are usually absent (although I'm certainly not claiming that they don't play a role). The reason being is that conformal field theories have an entirely different approach, which is based on symmetries and integrability. For instance: for a lot of conformal field theory you only care about the representation theory of the Hilbert space -- you don't even need to know the action S of the theory! This is very peculiar if you are used to doing everything via perturbation theory.

With that being said, you still need to know a great deal about the basic stuff of QFT. Chapter 2 of DiFrancesco precisely covers it, but it might be a bit compact. My advise would be: start with that chapter to familiarise with the necessary topics, then move on to other books and cover the same topics again. Don't venture off to deeply on the topics I mentioned above -- you won't need them. Focus on: what is a field, what is a correlator and what roles do symmetries play in QFT. Get to know the free / interacting boson theory, but skip stuff like loop diagrams and renormalization. Don't worry too much about trying to comprehend functional integration at first, although you will need it later on when you learn about modular invariance.

Other books that might appeal to you are:
Kac -- Vertex Algebras for beginners
Ralph Blumenhagen and Erik Plauschinn -- Introduction to Conformal Field Theory: With Applications to String Theory

 

[Tom Gilroy]

Do not go through the entire book of Srednicki if your primary goal is CFT. The reason is that QFT in the conventional way has some aspects which are fairly absent in CFT. For instance:

-perturbation theory
-renormalization
-running coupling constants
-gauge theory
-symmetry breaking

are usually large topics that need to be tackled in a course on QFT. But for a first treatment on CFT these are usually absent (although I'm certainly not claiming that they don't play a role). The reason being is that conformal field theories have an entirely different approach, which is based on symmetries and integrability. For instance: for a lot of conformal field theory you only care about the representation theory of the Hilbert space -- you don't even need to know the action S of the theory! This is very peculiar if you are used to doing everything via perturbation theory.

Thank you very much for the excellent response! As I mentioned, it's not so much QFT for the sake of QFT that I'm interested in, I'm more interested in developing the prerequisite QFT for CFT. It's really helpful to know some of the topics in QFT that are of less immediate importance for my intentions.

With that being said, you still need to know a great deal about the basic stuff of QFT. Chapter 2 of DiFrancesco precisely covers it, but it might be a bit compact. My advise would be: start with that chapter to familiarise with the necessary topics, then move on to other books and cover the same topics again. Don't venture off to deeply on the topics I mentioned above -- you won't need them. Focus on: what is a field, what is a correlator and what roles do symmetries play in QFT. Get to know the free / interacting boson theory, but skip stuff like loop diagrams and renormalization. Don't worry too much about trying to comprehend functional integration at first, although you will need it later on when you learn about modular invariance.

Ok, I'll start with that! Modular invariance is pretty central to my interests, so I'll probably need the functional integration a little sooner I'd guess.

Other books that might appeal to you are:
Kac -- Vertex Algebras for beginners

As I mentioned in the original post, I'm actually studying Vertex Operator Algebras for my Ph.D, so I'm very familiar with Kac's book. The main reason I'd like to learn CFT is because algebraically, they're essentially special cases of VOAs, and I feel that developing a second, different perspective would allow me to develop and deepen my understanding of these structures further.

Ralph Blumenhagen and Erik Plauschinn -- Introduction to Conformal Field Theory: With Applications to String Theory

I haven't heard of this book. I'll be sure to look into it.

 

[xepma]

Well, if you know Kac's book and actually understand the majority of that book, then you really should just move on to the book by DiFrancesco (et al) or the other CFT books (Blumenhagen and Plauschinn, or the book Conformal Field Theory by Ketov, or the article by Ginsparg, take your pick). Don't bother too much with the regular books on QFT.

 

[Tom Gilroy]

Thanks. I'll just dive right in I guess!

 

[Kevin_Axion]

I'd like to point out that the Conformal Field Theory Lectures began today at PSI.

 

[A. Neumaier]

For QFT you may want to take a look at Folland's "Quantum Field Theory. A Tourist Guide for Mathematicians". Written by a mathematician for mathematicians.

Zeidler's two volumes on QFT http://www.mis.mpg.de/zeidler/qft.html
are very suitable with your math background! One doesn't need to read it in order - just pick what attracts your interest.

 

[Landau]

I see A Mathematical Introduction to Conformal Field Theory hasn't been mentioned yet.