Conformal field theory on the plane

TL;DR

This work presents a comprehensive bootstrap-based treatment of two-dimensional conformal field theory on the plane, emphasizing symmetry and consistency over Lagrangian methods. It builds from the Virasoro algebra and Ward identities to conformal blocks, then leverages degenerate fields and BPZ equations to obtain tractable blocks and crossing relations. The text then applies these tools to Liouville theory and generalized/minimal models, deriving the DOZZ three-point constant, fusion rules, and Kac table structure, and discusses limits and interrelations among models. Altogether, the approach yields a cohesive picture where non-rational and rational 2D CFTs are understood through universal blocks, fusion algebra, and crossing symmetry, with practical formulas for correlation functions and structure constants. The methods illuminate how central charge, conformal dimensions, and extended symmetries constrain the spectrum and enable exact solutions in many important theories.

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.