Three-point functions in critical loop models

In two-dimensional models of critical non-intersecting loops, there are $\ell$-leg fields that insert $\ell∈\mathbb{N}^*$ open loop segments, and diagonal fields that change the weights of closed loops. We conjecture an exact formula for 3-point functions of such fields on the sphere. In the cases of diagonal or spinless 2-leg fields, the conjecture agrees with known results from Conformal Loop Ensembles. We numerically compute 3-point functions in loop models on cylindrical lattices, using transfer matrix techniques. The results agree with the conjecture in almost all cases. We attribute the few discrepancies to difficulties that can arise in our lattice computation when the relevant modules of the unoriented Jones-Temperley–Lieb algebra have degenerate ground states.