Conformal field theory on the plane

TL;DR

This paper reviews conformal field theory on the plane using the bootstrap approach, covering mathematical structures, key models, and their interrelations in two-dimensional theories with local conformal symmetry.

Contribution

It provides a comprehensive overview of the conformal bootstrap method applied to 2D CFTs, detailing mathematical tools and key models like Liouville and WZW theories.

Findings

Detailed description of Virasoro algebra and conformal blocks

Analysis of relations and limits among key models

Application of bootstrap approach to various 2D CFTs

Abstract

We review conformal field theory on the plane in the conformal bootstrap approach. We introduce the main ideas of the bootstrap approach to quantum field theory, and how they apply to two-dimensional theories with local conformal symmetry. We describe the mathematical structures that appear in such theories, from the Virasoro algebra and its representations, to BPZ equations and conformal blocks. Examples include Liouville theory, (generalized) minimal models, free bosonic theories, the $H_3^+$ model, and the $SU_2$ and $\widetilde{SL}_2(\mathbb{R})$ WZW models. We also discuss relations between some of these models, and limits of these models when the central charge and/or conformal dimensions tend to particular values.