Exactly Solvable 2D Conformal Field Theory - Lecture 1: The Virasoro Algebra and Its Representations

Embark on a comprehensive exploration of two-dimensional Conformal Field Theory (CFT) in this first lecture of a six-part series. Delve into the bootstrap approach, focusing on exactly solvable CFTs without extended chiral symmetry. Learn about the Virasoro algebra and its representations, as well as fields and operator product expansions. Discover how local conformal symmetry and degenerate fields constrain the spectrum and correlation functions. Examine the crossing symmetry equations and their analytical and numerical solutions, leading to analytic formulas for structure constants. Explore Liouville theory, (generalized) minimal models, and loop models including O(n), Potts, and U(n) models. Gain insights into the combinatorial description of correlation functions in loop models, inspired by lattice constructions of statistical models. This two-hour lecture sets the foundation for understanding exactly solvable 2D CFTs and outlines the challenges that remain in solving these models.