Minimal lectures on two-dimensional conformal field theory

TL;DR

This paper offers a concise, self-contained review of two-dimensional conformal field theory, covering fundamental principles, models, and the bootstrap approach, with detailed analysis of minimal models and Liouville theory.

Contribution

It provides a clear, accessible overview of 2D CFT, including derivations of key equations and detailed solutions for minimal models and Liouville theory.

Findings

Derived Ward identities and BPZ equations from Virasoro algebra

Solved minimal models and Liouville theory correlation functions

Discussed existence and uniqueness of conformal blocks

Abstract

We provide a brief but self-contained review of two-dimensional conformal field theory, from the basic principles to some of the simplest models. From the representations of the Virasoro algebra on the one hand, and the state-field correspondence on the other hand, we deduce Ward identities and Belavin--Polyakov--Zamolodchikov equations for correlation functions. We then explain the principles of the conformal bootstrap method, and introduce conformal blocks. This allows us to define and solve minimal models and Liouville theory. In particular, we study their three- and four-point functions, and discuss their existence and uniqueness. In appendices, we introduce the free boson theory (with an arbitrary central charge), and the modular bootstrap in minimal models.