Superintegrability of $d$-dimensional Conformal Blocks

TL;DR

This paper reveals that conformal blocks in any dimension can be understood as eigenfunctions of a superintegrable Calogero-Sutherland Hamiltonian, linking conformal field theory to integrable models and enabling new analytical tools.

Contribution

It establishes a novel mapping between conformal blocks and Calogero-Sutherland models, allowing explicit construction and duality relations across dimensions.

Findings

Explicit construction of conformal blocks using Heckman-Opdam functions

Discovery of a duality relating blocks in different dimensions

Connection of conformal blocks to superintegrable quantum systems

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.