Superintegrability of $d$-Dimensional Conformal Blocks

We observe that conformal blocks of scalar four-point functions in a $d$-dimensional conformal fieldtheory can be mapped to eigenfunctions of a two-particle hyperbolic Calogero-Sutherland Hamiltonian.The latter describes two coupled Pöschl-Teller particles. Their interaction, whose strength dependssmoothly on the dimension $d$, is known to be superintegrable. Our observation enables us to exploit the richmathematical literature on Calogero-Sutherland models in deriving various results for conformal fieldtheory. These include an explicit construction of conformal blocks in terms of Heckman-Opdamhypergeometric functions. We conclude with a short outlook, in particular, on the consequences ofintegrability for the theory of conformal blocks