Singular Vectors in Liouville CFT: Representation Theory Insights

This lecture discusses singular vectors in the context of Liouville Conformal Field Theory (CFT). The instructor begins by recalling concepts from previous lectures, particularly focusing on the representation theory associated with singular vectors. They explain the significance of singular vectors as axiomatic elements in physics, particularly in the study of representations of the Viasso algebra. The lecture delves into the mathematical framework, including the Hilbert space structure and the Hamiltonian involved in the analysis. The instructor highlights the probabilistic approach to Liouville theory, emphasizing the role of Gaussian fields and their properties. They also discuss the implications of the catch table on the linear independence of singular vectors and the conditions under which they vanish. The lecture concludes with applications of these concepts to Riemann surfaces and the classification of modules, providing a comprehensive overview of the theoretical underpinnings of singular vectors in Liouville CFT.