Correlation Functions in Loop Models: Structural Assumptions

This lecture discusses correlation functions in statistical loop models, focusing on the characterization of primary fields by their left and right momenta. The instructor introduces the concept of conformal dimensions, explaining the relationship between conformal dimensions and momenta. The significance of degenerate fields, particularly the unique degenerate field V12, is emphasized as a crucial structural assumption in conformal field theory (CFT). The implications of this assumption on operator product expansions and the constraints it imposes on non-diagonal fields are explored. The instructor contrasts the situation in loop models with that in minimal models, highlighting the differences in the types of degenerate fields present. The lecture concludes with a summary of the qualitative features of loop models compared to UV theory, noting the distinct nature of correlation functions and the solvability of loop models. The discussion includes insights into the mathematical structures underlying these theories and their implications for understanding critical phenomena in statistical mechanics.