My master research topic is about developing new numerical methods (see our paper) to analyze critical phenomena; the conformal field theory (CFT) is one of the most important languages to describe critical phenomena. Although I realized that I should start learning CFT in early 2020, it was not until quite recent that I actually start. I’ve scanned the following lecture notes and videos during the past two weeks:
- EPFL lectures on Conformal Field Theory in $D \leq 3$ Dimensions, by Slava Rychkov. Here you can find the arXiv version and the published Springer version. I choose to first read this notes because now I’m extracting scaling dimensions of the 3D Ising model.
- Five lecture videos by John Cardy talking about conformal field theory and statistical mechanics. Here is the YouTube link. Cardy focused on 2d CFT results, and he also mentioned many interesting statistical models like the simplest Gaussian, height model, and loop gas model.
EPFL lecture notes
This short note is less than 100 pages and discusses the form of 2- and 3-point functions and also radial quantization. The focus is on general spatial dimension $D>=3$ instead of $D=2$.
The most interesting thing about this lecture note is how it addresses conformal transformation and explains conformal kinematics. Conformal is nothing but adding scaling (zooming in and out) and the so-called special conformal transformation into the ordinary rotation group $SO(D)$ in $D-$spatial dimension. This means there are no free-offered $D$-dimensional defining representation for the conformal group because the map from the old coordinate $\vec{x}$ to the new $\vec{x’}$ is nonlinear. However, interestingly, the conformal group in $D-$spatial dimension is the same as, or if you want a more mathematical term, isomorphic to, the group $SO(D+1, 1)$, the Lorentz group in $(D+1) + 1$-space-time dimension. Since we know very well about how the group $SO(D+1,1)$ acts on a $(D+2)$-dimensional vector $\vec{X}$, if we can find a way to embed the $D$-dimensional vector $\vec{x}$ into the higher dimensional space $\vec{X}$, we can study the conformal group by thinking about $SO(D+1, D)$. The submanifold embedded in $\vec{X}$ is called the Projective Null Cone in the book. This is an interesting trick that most textbooks in CFT do not mention.
Cardy's lecture videos
Cardy’s lecture is short and extremely enlightening! The interesting thing about his lecture is that he talks many fascinating lattice models and also briefly explain how to identify them with the known 2d CFTs. I will definitely read his lecture notes carefully.
Tentative plans
To better understand CFT in general spacial dimensions larger than 2, I also want to take a look at David Simmons-Duffin’s TASI lecture notes. I think it is closely related to the EPFL notes above, but it contains many interesting short exercises.
For the 2D CFT, apart from Cardy’s lecture notes and his textbook, I will also resume reading the big yellow book, where many detailed calculations are shown.
