Superintegrability of $d$-dimensional Conformal Blocks

TL;DR

This work reveals that the Casimir equations governing $d$-dimensional scalar conformal blocks are equivalent to the Schrödinger problem for a BC$_2$ Calogero-Sutherland system, with the dimension acting as a continuous coupling $\varepsilon=d-2$. By connecting conformal blocks to Heckman–Opdam hypergeometric functions and exploiting the superintegrable structure, the authors derive a gauge-linked two-particle PT system in $d=2$ and its fully coupled extension for general $d$, along with a rich web of relations to DAHA, KOornwinder polynomials, and RS duality. This framework yields a duality that maps blocks across dimensions and provides a route to analytic expressions for crossing kernels and bootstrap data. The approach bridges conformal field theory with integrable systems and special function theory, offering new tools for the conformal bootstrap and potential extensions to superconformal theories.

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 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 and a remarkable duality that relates the blocks of theories in different dimensions.