Exact solution of three-point functions in critical loop models

We propose an exact formula for three-point functions on the sphere in critical loop models with primary fields $V_{(r,s)}$ characterized by $2r$ legs and a parameter $s$ that describes diagonal fields for $r=0$ and the momentum of legs for $r>0$. We demonstrate its validity in three ways: the conformal bootstrap method for 4-point functions, a transfer-matrix study of the lattice model, and a probabilistic method based on conformal loop ensemble and Liouville quantum gravity. This work provides a crucial missing piece for solving critical loop models and reveals a deep unity between three fundamental approaches to 2D statistical physics: transfer matrix, conformal field theory, and probability theory.