An Introduction to Conformal Field Theory

TL;DR

This work presents a comprehensive algebraic framework for two-dimensional conformal field theories, focusing on meromorphic (chiral) theories and their extension to full local theories. It builds from the space of states and meromorphic correlation functions to the vertex-operator algebra structure, using factorisation, OPEs, and the Virasoro algebra to organize representations. Central contributions include Zhu's algebra for classifying representations, the fusion-algebra framework with the Verlinde formula, and the Moore–Seiberg bootstrap that ties fusion and braiding to modular data, complemented by concrete examples from free bosons, affine theories, lattice models, cosets, and orbifolds. The paper also surveys extensions to logarithmic theories, supersymmetry, and connections to quantum groups, highlighting both the powerful predictive structure and outstanding challenges in higher-genus and non-rational settings. Together, these insights underpin the exact solvability of a broad class of CFTs and their deep interplay with mathematics and string theory.

Abstract

A comprehensive introduction to two-dimensional conformal field theory is given.