Torus one-point functions in critical loop models

We show that in critical loop models, torus 1-point functions can be expressed in terms of sphere 4-point functions at a different central charge. Unlike in the Moore--Seiberg formalism, crossing symmetry on the sphere therefore implies modular covariance on the torus. We systematically compute torus 1-point functions in critical loop models, using a numerical bootstrap approach. We focus on the 1-point functions of the 6 simplest primary fields, which give rise to 10 solutions of modular covariance equations. Such 1-point functions are infinite linear combinations of conformal blocks. The coefficients are products of double Gamma functions, times polynomial functions of loop weights. For each solution, we determine the first 6 to 12 polynomials.