Foundations of Liouville Field Theory

How do geometry and quantum physics merge to describe the fabric of two-dimensional space? Liouville field theory provides the key, linking the classical geometry of Riemann surfaces with the quantum world of string theory and 2D gravity. This text-only course guides you from the absolute basics of classical surface geometry to the complex structures of quantum conformal field theory. By reading our structured explanations and working through analytical exercises, you will build a solid intuitive and mathematical grasp of this profound area of mathematical physics. You will: Understand the classical Liouville equation and its historical role in the uniformization of Riemann surfaces; Explore how quantum Liouville theory arises as a conformal anomaly in bosonic string theory; Learn the fundamentals of conformal field theory, including Virasoro algebra and primary fields; Analyze the modern probabilistic formulation of Liouville theory using Gaussian Free Fields; Practice calculating conformal weights and solving field equations through step-by-step written derivations. The course begins with essential definitions of complex manifolds and classical fields, ensuring you have a strong foundation. You will then progress through quantum path integrals, the bootstrap program, and contemporary mathematical physics applications. Designed for students, researchers, and physics enthusiasts looking for an accessible, beginner-friendly entry point into advanced mathematical physics, a basic background in calculus and introductory quantum mechanics is helpful, but no prior knowledge of string theory is required. Start exploring the mathematical foundations of quantum gravity today.