Conformal group theory of tensor structures

• Published in: The journal of high energy physics 2020-10, Vol.2020 (10), p.1-39, Article 4
• Main Authors: Burić, Ilija, Schomerus, Volker, Isachenkov, Mikhail
• Format: Article
• Language: English
• Subjects: Classical and Quantum Gravitation; Conformal Field Theory; Correlation; Correlators; Decomposition; Elementary Particles; Euclidean geometry; Euclidean space; Field theory; Global Symmetries; Group theory; High energy physics; High Energy Physics - Theory; Mathematical models; Operators (mathematics); Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; Space-Time Symmetries; String Theory; Tensors; Wave functions
• ISSN: 1029-8479; 1126-6708; 1029-8479
• DOI: 10.1007/JHEP10(2020)004

A bstract 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.