Statistical Models: Unitary Conformal Field Theories

This lecture discusses the relationship between statistical models and unitary conformal field theories (CFTs). The instructor begins by introducing statistical models at critical points, particularly focusing on the Ising model. The discussion emphasizes the importance of local lattice fields and their convergence to continuous fields in CFT. The instructor explains how scaling dimensions can be observed in limiting theories and how correlations behave under various transformations. The lecture also covers the significance of the stress tensor in CFT, including its holomorphic properties and the implications of the Virasoro algebra. The instructor highlights the role of unitarity conditions and the Shapovalov form in ensuring positive semi-definiteness of correlation matrices. Throughout the lecture, the instructor provides insights into the mathematical underpinnings of these theories, including the Kac determinant formula and its relevance to understanding the behavior of fields in CFT. The lecture concludes with a discussion on the characterization of unitary conformal families and the conditions under which they hold.