Superintegrability of d-Dimensional Conformal Blocks.

Abstract

We observe that conformal blocks of scalar four-point functions in a d-dimensional conformal field theory 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 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. We conclude with a short outlook, in particular, on the consequences of integrability for the theory of conformal blocks.