Conformal Group Theory of Tensor Structures

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.

References in corpus (22)

• Bounding scalar operator dimensions in 4D CFT
• Solving the 3D Ising Model with the Conformal Bootstrap
• Conformal Four Point Functions and the Operator Product Expansion
• Conformal Partial Waves and the Operator Product Expansion
• Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents
• Precision Islands in the Ising and $O(N)$ Models
• Projectors, Shadows, and Conformal Blocks
• Spinning Conformal Blocks
• Radial Coordinates for Conformal Blocks
• Weight Shifting Operators and Conformal Blocks
• Superintegrability of $d$-dimensional Conformal Blocks
• Conformal correlators of mixed-symmetry tensors
• Projectors and seed conformal blocks for traceless mixed-symmetry tensors
• Radial expansion for spinning conformal blocks
• Seed Conformal Blocks in 4D CFT
• Harmony of Spinning Conformal Blocks
• Spinning operators and defects in conformal field theory
• From Spinning Conformal Blocks to Matrix Calogero-Sutherland Models
• Integrability of Conformal Blocks I: Calogero-Sutherland Scattering Theory
• Calogero-Sutherland Approach to Defect Blocks
• Spinning conformal defects
• Superconformal Blocks: General Theory

Cited by in corpus (8)

• Seven-Point Conformal Blocks in the Extended Snowflake Channel and Beyond
• Gaudin Models and Multipoint Conformal Blocks III: Comb channel coordinates and OPE factorisation
• Six-Point Conformal Blocks in the Snowflake Channel
• Towards Feynman rules for conformal blocks
• Defect Conformal Blocks from Appell Functions
• Universal Spinning Casimir Equations and Their Solutions
• Superconformal blocks in diverse dimensions and $BC$ symmetric functions
• Crossing Symmetry for Long Multiplets in 4D $\mathcal{N}=1$ SCFTs