Superintegrability of $d$-dimensional Conformal Blocks

Abstract

We observe that conformal blocks of scalar 4-point functions in a $d$-dimensional conformal field theory can mapped to eigenfunctions of a 2-particle hyperbolic Calogero-Sutherland Hamiltonian. The latter describes two coupled P\"oschl-Teller particles. Their interaction, whose strength depends smoothly on the dimension $d$, is known to be superintegrable. Our observation enables us to exploit the rich mathematical literature on Calogero-Sutherland models in deriving various results for conformal field theory. These include an explicit construction of conformal blocks in terms of Heckman-Opdam hypergeometric functions and a remarkable duality that relates the blocks of theories in different dimensions.