Modern Approach to 2D Conformal Field Theory

TL;DR

This paper introduces modern analytical methods in 2D conformal field theory, emphasizing irrational CFTs and their relevance to quantum gravity via AdS/CFT, filling a gap left by traditional textbooks.

Contribution

It provides a comprehensive overview of recent techniques and results in irrational 2D CFTs, crucial for understanding quantum gravity and beyond textbook knowledge.

Findings

Development of HHLL block and monodromy method

Discussion of Hellerman and HKS bounds

Application of ICFT methods to quantum gravity

Abstract

The primary aim of these lecture notes is to introduce the modern approach to two-dimensional conformal field theory (2D CFT). The study of analytical methods in two-dimensional conformal field theory has developed over several decades, starting with BPZ. The development of analytical methods, particularly in rational conformal field theory (RCFT), has been remarkable, with complete classifications achieved for certain model groups. One motivation for studying CFT comes from its ability to describe quantum critical systems. Given that realistic quantum critical systems are fundamentally RCFTs, it is somewhat natural that the analytical methods of RCFT have evolved significantly. CFTs other than RCFTs are called irrational conformal field theories (ICFTs). Compared to RCFTs, the study of ICFTs has not progressed as much. Leaving aside whether there is physical motivation or not, ICFTs inherently possess a difficulty that makes them challenging to approach. However, with the development of quantum gravity, the advancement of analytical methods for ICFTs has become essential. The reason lies in the AdS/CFT correspondence. AdS/CFT refers to the relationship between $d+1$ dimensional quantum gravity and $d$ dimensional CFT. Within this correspondence, the CFT appears as a non-perturbative formulation of quantum gravity. Except in special cases, this CFT belongs to ICFT. Against this backdrop, the methods for ICFTs have developed rapidly in recent years. Many of these ICFT methods are indispensable for modern quantum gravity research. Unfortunately, these cannot be learned from textbooks on 2D CFTs, such as Yellow book. These lecture notes aim to fill this gap. Specifically, we will cover techniques that have already been applied in many studies, such as HHLL block and monodromy method, and significant results that have become proper nouns, such as Hellerman bound and HKS bound.