Conformal Loop Ensembles: Connections to CFT

This lecture discusses the relationship between conformal loop ensembles (CLE) and conformal field theory (CFT), focusing on the three-point nesting function for CLE. The instructor introduces Schramm-Loewner evolution (SLE) as a fundamental concept, explaining its properties and how it serves as a scaling limit for various statistical physics interfaces, such as the critical Ising model and percolation. The lecture emphasizes the conformal invariance of SLE and its significance in understanding lattice models at criticality. The instructor then transitions to CLE, describing it as a random collection of non-crossing loops that arise from the scaling limits of lattice model interfaces. The discussion includes deep predictions from physics literature regarding the connection between CLE observables and CFT structure constants, particularly the imaginary DOZZ formula. The goal of the mini-course is to rigorously derive the relationship between the CLE three-point nesting function and the imaginary DOZZ formula, highlighting the mathematical underpinnings of these concepts.