Unitary Minimal Models: Representation Theory Insights

This lecture discusses the representation theory of unitary minimal models, focusing on the Virasoro algebra. The instructor begins by outlining the importance of representation theory as a foundational element for minimal models, emphasizing the role of conformal field theories. The discussion includes the construction of Hilbert spaces associated with circles and the decomposition into irreducible representations of the Virasoro algebra. The lecture also covers modular invariance and the partition function of the torus, which are crucial for understanding the gluing of irreps. The instructor explains the significance of the highest weight modules and the Verma modules, detailing how they are constructed and their unique properties. Throughout the lecture, the instructor addresses questions regarding the physical motivations behind the assumptions made, particularly concerning the self-adjointness and diagonalizability of operators. The session concludes with a discussion on the characterization of irreducible representations and the role of singular vectors in this context, providing a comprehensive overview of the theoretical framework underpinning unitary minimal models.