  We establish that all of the one- and two-dimensional global conformal blocks
are, up to some choice of prefactor, free-particle wavefunctions in tensor
products of AdS$_3$ or limits thereof. Our first core observation is that the
six-point comb-channel conformal blocks correspond to free-particle
wavefunctions on an AdS$_3$ constructed directly in cross-ratio space. This
construction generalizes to blocks for a special class of diagrams, which are
determined as free-particle wavefunctions in tensor products of AdS$_3$.
Conformal blocks for all the remaining topologies are obtained as limits of the
free wavefunctions mentioned above. Our results show directly that the
integrable models associated with all one- and two-dimensional conformal blocks
can be seen as limits of free theory, and manifest a relation between AdS and
CFT kinematics that lies outside of the standard AdS/CFT dictionary. We
complete the discussion by providing explicit Feynman-like rules that can be
used to work out blocks for all topologies, as well as a Mathematica notebook
that allows simple computation of Casimir equations and series expansions for
blocks, by requiring just an OPE diagram as input.
