Harmony of Spinning Conformal Blocks

TL;DR

This work develops a universal framework for spinning conformal blocks by harmonic analysis on vector bundles over a conformal coset, recasting the Casimir equations as a matrix Calogero-Sutherland Schrödinger problem for two interacting spinning particles in a 1D potential. It systematically constructs spinning blocks from induced representations, reduces the Laplacian on the conformal group to a 2D Cartan subspace, and derives matrix CS Hamiltonians whose spectra encode spinning Casimir equations. The method is illustrated with explicit 3D seed blocks: scalar seeds reproduce the known scalar CS system, while fermionic seeds yield a 4×4 matrix Hamiltonian that matches established spinning Casimir equations (with spin-flip structure controlled by the external spins). The approach reveals a deep connection between conformal blocks and integrable quantum systems and suggests rich extensions to higher dimensions, defects, and supersymmetric contexts, with potential for algebraic solutions via super-integrability.

Abstract

Conformal blocks for correlation functions of tensor operators play an increasingly important role for the conformal bootstrap programme. We develop a universal approach to such spinning blocks through the harmonic analysis of certain bundles over a coset of the conformal group. The resulting Casimir equations are given by a matrix version of the Calogero-Sutherland Hamiltonian that describes the scattering of interacting spinning particles in a 1-dimensional external potential. The approach is illustrated in several examples including fermionic seed blocks in 3D CFT where they take a very simple form.