Conformal Loop Ensembles: Perspectives and Applications

This lecture focuses on conformal loop ensembles and their relationship with conformal field theory (CFT). It begins with a recap of Liouville CFT on a domain, discussing the Liouville field and its correlation functions. The instructor presents two perspectives on Liouville quantum gravity: the Liouville conformal field theory perspective and the random surface perspective, highlighting their equivalence in describing random geometry. The lecture further explores special quantum surfaces, particularly the quantum sphere, and discusses the laws governing these surfaces. The importance of correlation functions in boundary Liouville CFT is emphasized, along with their applications in computing observables in stochastic processes like Schramm-Loewner evolution (SLE). The instructor also addresses the equivalence of perspectives in practical applications, demonstrating how tools from both perspectives can be utilized to solve problems in Liouville CFT and random surfaces. The lecture concludes with examples of cutting LQG by chordal SLE and the implications of these methods in understanding quantum surfaces.