  We demonstrate how to obtain integrable results for the Schramm-Loewner
evolution (SLE) from Liouville conformal field theory (LCFT) and the
mating-of-trees framework for Liouville quantum gravity (LQG). In particular,
we prove an exact formula for a conformal derivative of a classical variant of
SLE called $\mathrm{SLE}_\kappa(\rho_-;\rho_+)$. Moreover, we prove that the
SLE loop measure constructed by Zhan (2020) arises naturally from the conformal
welding of two quantum disks in LQG. Our proof is built on two connections
between SLE, LCFT, and mating-of-trees. Firstly, LCFT and mating-of-trees
provide equivalent but complementary methods to describe natural random
surfaces in LQG. Secondly, conformal welding of these random surfaces produces
SLE curves as their interfaces. To prove our results, we provide a unified
approach to the first connection. Moreover, we rely on the conformal welding
results proved in our companion paper [AHS20]. Our paper is an essential part
of a program proving integrability results for SLE, LCFT, and mating-of-trees
based on these two connections.
