On a given Riemann surface, C. Guillarmou, A. Kupiainen and R. Rhodes constructed a path integral based on the Liouville action functional with imaginary parameters, called the Compactified Imaginary Liouville Theory (CILT). They proved that CILT is a conformal field theory (CFT) and satisfies Segal's axioms. In this talk I will go through their construction of CILT, with some motivation (coulomb gas& minimal models) and intuition (critical loop models). If time permitted, I will also talk about some general techiques of CFT ever since Belavin, Polyakov and Zamolodchikov, 1984 and how they can solve the conformal bootstrap program. This talk is also based on joint (ongoing) work with C. Guillarmou, A. Kupiainen, R. Rhodes and Y. Xie.
