Harmony of spinning conformal blocks

Volker Schomerus,Mikhail Isachenkov,Evgeny Sobko

doi:10.1007/jhep03(2017)085

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.

Highlights

The bootstrap programme, such formulas are difficult to work with, partly because they involve a the large number of integrations
We develop a universal approach to such spinning blocks through the harmonic analysis of certain bundles over a coset of the conformal group
Throughout the two sections we shall set up a model for spinning conformal blocks in any dimension where the 4-point blocks are represented as sections in a certain vector bundle over the following double coset of the conformal group G = SO(1, d + 1)

Summary

Conformal blocks and tensor structures

We shall review the basic model of spinning conformal blocks in the context of 4-point correlation functions on Rd. Correlation function of three primary operators corresponding to representations (∆1, μ1), (∆2, μ2) and (∆3, μ3) can be written as a sum over conformally invariant tensor structures tα. As in our analysis of 3-point structures, we obtain an alternative view on the tensor structures if we evaluate 4-point correlation functions by performing operator product expansion of two fields O1 and O2 into conformal primary fields O = Oπ and its descendants. Where C(2) denotes the second order Casimir differential operator and C∆,μ is the eigenvalue of the quadratic Casimir element of the conformal group in the representation χπ that is induced from (∆, μ) Such Casimir equations are well known for scalar blocks, see [20], and they were constructed for several examples involving fields with spin, see [18, 21]. Our main goal in this work is to develop a systematic approach to Casimir equations for spinning blocks

Harmonic analysis approach to conformal blocks

Harmonic analysis and Calogero-Sutherland models

Cartan decomposition of the conformal group

The Laplacian on the Cartan subgroup

Example: seed conformal blocks in 3D

Discussion, outlook and conclusions