Quantum integrable systems from conformal blocks

In this note, we extend the striking connections between quantum integrable systems and conformal blocks recently found in [M. Isachenkov and V. Schomerus, Phys. Rev. Lett. 117, 071602 (2016)] in several directions. First, we explicitly demonstrate that the action of the quartic conformal Casimir operator on general d-dimensional scalar conformal blocks can be expressed in terms of certain combinations of commuting integrals of motions of the two particle hyperbolic BC 2 Calogero-Sutherland system. The permutation and reflection properties of the underlying Dunkl operators play crucial roles in establishing such a connection. Next, we show that the scalar superconformal blocks in superconformal field theories (SCFTs) with four and eight supercharges and suitable chirality constraints can also be identified with the eigenfunctions of the same Calogero-Sutherland system; this demonstrates the universality of such a connection. Finally, we observe that the so-called "seed" conformal blocks for constructing four point functions for operators with arbitrary space-time spins in four-dimensional CFTs can also be linearly expanded in terms of Calogero-Sutherland eigenfunctions.