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

One- and two-dimensional higher-point conformal blocks as free-particle wavefunctions in AdS$_3^{\otimes m}$