Local Fields and Correlations in Ising Models

This lecture discusses the concept of local fields and correlations within the context of Ising models and conformal field theory (CFT). The instructor begins by reviewing the Ising model, emphasizing its random configurations and boundary conditions. The discussion transitions to the discrete Gaussian free field, comparing it with the Ising model to motivate conjectures about local fields. The instructor introduces the notion of local fields as locally defined random variables, providing examples from both the Ising model and the discrete Gaussian free field. The lecture highlights the significance of local fields in CFT, particularly their correspondence to the state space of the theory. The instructor outlines three key dreams regarding the relationship between local fields in lattice models and their counterparts in CFT, including the existence of commuting actions of the Virasoro algebra. The lecture concludes with a discussion of the challenges and conjectures related to these dreams, emphasizing the importance of understanding local fields in both probabilistic and field-theoretic contexts.