Gauge Theories: Chiral Transformations and Path Integrals

This lecture discusses the gauged Wess-Zumino-Witten (WZW) model, focusing on its gauge symmetry and the introduction of gauge fields. The instructor defines the action and correlation functions, emphasizing the role of chiral gauge transformations. The lecture also covers the Ward identity and the Polyakov-Wiegman identity, which relate to the partition function and path integrals. The concept of holomorphic factorization is introduced, linking Kac-Moody blocks to the state space of 3D Chern-Simons quantum field theory. The discussion extends to the sigma model on hyperbolic space, particularly for the group SU(2), and its probabilistic formulation in terms of Gaussian multiplicative chaos. The instructor explains iterated Gaussian integrals and the renormalization process necessary for defining the path integral. The lecture concludes with insights into the connection between the WZW model and Liouville theory, highlighting the significance of correlation functions and their implications in conformal field theory.