Quantum Integrable Systems from Conformal Blocks

TL;DR

This paper reveals deep connections between conformal blocks in various supersymmetric and non-supersymmetric conformal field theories and quantum integrable systems, specifically the Calogero-Sutherland model, extending previous findings.

Contribution

It explicitly demonstrates the link between conformal Casimir operators and Calogero-Sutherland integrals, showing universality across different types of conformal blocks.

Findings

Conformal Casimir acts as Calogero-Sutherland Hamiltonian.

Superconformal blocks are eigenfunctions of the same integrable system.

Seed conformal blocks can be expanded in Calogero-Sutherland eigenfunctions.

Abstract

In this note, we extend the striking connections between quantum integrable systems and conformal blocks recently found in http://arxiv.org/abs/1602.01858 in several directions. First, we explicitly demonstrate that the action of 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 BC2 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 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.