Harmony of Spinning Conformal Blocks

TL;DR

This paper introduces a universal harmonic analysis method for spinning conformal blocks, leading to matrix Casimir equations akin to Calogero-Sutherland models, with applications to 3D fermionic seed blocks.

Contribution

It develops a novel, universal approach to spinning conformal blocks using harmonic analysis on cosets, resulting in matrix Casimir equations for the conformal bootstrap.

Findings

Casimir equations expressed as matrix Calogero-Sutherland Hamiltonians

Simplified form of fermionic seed blocks in 3D CFT

Demonstrated applicability to various examples

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.