Conformal field theory: a case study

TL;DR

The work presents a detailed, non-perturbative treatment of the Wess-Zumino-Witten model as a cornerstone of two-dimensional conformal field theory, connecting sigma-model dynamics with a topological WZ term to affine Kac-Moody algebras and the Virasoro algebra. It develops the exact solution via functional integrals, harmonic analysis on groups, and the Chern-Simons/WZW correspondence, deriving the Verlinde fusion rules, modular properties, and holomorphic factorization of correlation functions. The formalism extends to coset constructions and boundary CFT, including Ishibashi and Cardy states, gluing relations, and open/closed string amplitudes, illustrating the deep interplay between geometry, representation theory, and quantum field theory. Together, these results provide a coherent framework for exact, finite-dimensional computations of amplitudes, correlation functions, and spectra in rational CFTs arising from WZW data, with implications for string theory and topological invariants. The formalism emphasizes the central role of the level $k$, modular invariance, and the Knizhnik-Zamolodchikov connection in organizing conformal blocks and their monodromies across genus and boundary conditions.

Abstract

This is a set of introductory lecture notes devoted to the Wess-Zumino-Witten model of two-dimensional conformal field theory. We review the construction of the exact solution of the model from the functional integral point of view. The boundary version of the theory is also briefly discussed.