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This lecture is the fourth part of a series on Probabilistic Conformal Field Theory by Antti Kupiainen at the Hausdorff Center for Mathematics. Explore the mathematical foundations of Conformal Field Theories (CFTs), which describe universal behavior of physical systems at second order phase transition points and are crucial in Quantum Field Theories. Learn about the probabilistic approach to CFT based on path integral formulation and its connection to Graeme Segal's geometric approach to conformal bootstrap. Discover two prominent CFTs: Liouville CFT, central to Liouville Quantum Gravity and random surfaces theory, and Wess-Zumino-Witten models with their rich representation theoretical content. The lecture addresses the ongoing debate about rigorous mathematical foundations of these theories, which have had profound impacts on representation theory and geometry since their structure was uncovered by Belavin, Polyakov, and Zamolodchicov in 1983.
