Conformal Group Theory of Tensor Structures

TL;DR

This paper introduces a group theoretic method based on conformal group decomposition to systematically construct tensor structures for conformal correlators, enhancing the Calogero-Sutherland approach for four-point functions.

Contribution

It develops a universal, group-theoretic formula for tensor structures in conformal field theory, enabling systematic derivation of crossing equations and completing the Calogero-Sutherland framework.

Findings

Provides a new formula for tensor structures using Cartan decomposition.

Systematically derives crossing equations for conformal correlators.

Completes the Calogero-Sutherland approach for four-point functions.

Abstract

The decomposition of correlation functions into conformal blocks is an indispensable tool in conformal field theory. For spinning correlators, non-trivial tensor structures are needed to mediate between the conformal blocks, which are functions of cross ratios only, and the correlation functions that depend on insertion points in the $d$-dimensional Euclidean space. Here we develop an entirely group theoretic approach to tensor structures, based on the Cartan decomposition of the conformal group. It provides us with a new universal formula for tensor structures and thereby a systematic derivation of crossing equations. Our approach applies to a `gauge' in which the conformal blocks are wave functions of Calogero-Sutherland models rather than solutions of the more standard Casimir equations. Through this ab initio construction of tensor structures we complete the Calogero-Sutherland approach to conformal correlators, at least for four-point functions of local operators in non-supersymmetric models. An extension to defects and superconformal symmetry is possible.