Résumé: In critical limits of statistical models, non-local observables such as cluster connectivities can be conformally invariant without necessarily belonging to a conformal field theory. We propose to describe such observables in terms of conformal correlator systems, defined by weaker axioms than those of CFT. In particular, N-point functions can be reduced to 4-point functions, instead of 3-point functions in CFT.

In two dimensions, the spectrum of a conformal correlator system is not constrained by modular invariance. This allows the spin to take arbitrary complex values, leading to correlators with nontrivial monodromies. We show that sphere 4-point functions and torus 1-point functions are subject to nontrivial monodromy constraints. We give examples of correlators with abelian monodromies: explicit examples from free fields, and more complicated examples from critical loop models.