New Methods for Conformal Correlation Functions

Abstract

The most general operator product expansion in conformal field theory is obtained using the embedding space formalism and a new uplift for general quasi-primary operators. The uplift introduced here, based on quasi-primary operators with spinor indices only and standard projection operators, allows a unified treatment of all quasi-primary operators irrespective of their Lorentz group irreducible representations. This unified treatment works at the level of the operator product expansion and hence applies to all correlation functions. A very useful differential operator appearing in the operator product expansion is established and its action on appropriate products of embedding space coordinates is explicitly computed. This computation leads to tensorial generalizations of the usual Exton function for all correlation functions. Several important identities and contiguous relations are also demonstrated for these new tensorial functions. From the operator product expansion all correlation functions for all quasi-primary operators, irrespective of their Lorentz group irreducible representations, can be computed recursively in a systematic way. The resulting answer can be expressed in terms of tensor structures that carry all the Lorentz group information and linear combinations of the new tensorial functions. Finally, a summary of the well-defined rules allowing the computation of all correlation functions constructively is presented.

1+89 pages, 2 figures, extended discussion in Section 4, streamlined notation in Sections 5 and 6, extra identities in Appendix B, added references

References in corpus (14)

• Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents
• Bootstrapping Mixed Correlators in the 3D Ising Model
• Bounds on 4D Conformal and Superconformal Field Theories
• The $\mathcal{N}=8$ Superconformal Bootstrap in Three Dimensions
• Five dimensional $O(N)$-symmetric CFTs from conformal bootstrap
• Multipoint Conformal Blocks in the Comb Channel
• Harmonic Analysis and Mean Field Theory
• Recursion Relations in Witten Diagrams and Conformal Partial Waves
• Bootstrapping Mixed Correlators in Three-Dimensional Cubic Theories
• Exact Correlation Functions in Conformal Fishnet Theory
• Bounds on OPE Coefficients in 4D Conformal Field Theories
• Generalized Fishnets and Exact Four-Point Correlators in Chiral CFT${}_4$
• Bootstrapping MN and Tetragonal CFTs in Three Dimensions
• ${\varphi }^3$ theory with $F_4$ flavor symmetry in $6-2\epsilon$ dimensions: 3-loop renormalization and conformal bootstrap

Cited by in corpus (21)

• Gaudin Models and Multipoint Conformal Blocks: General Theory
• From Gaudin Integrable Models to $d$-dimensional Multipoint Conformal Blocks
• Higher-Point Conformal Blocks in the Comb Channel
• Conformal Primary Basis for Dirac Spinors
• Gaudin Models and Multipoint Conformal Blocks II: Comb channel vertices in 3D and 4D
• Recursion relations for 5-point conformal blocks
• 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
• More on holographic correlators: Twisted and dimensionally reduced structures
• blocks_3d: Software for general 3d conformal blocks
• Improving the five-point bootstrap
• Efficient Rules for All Conformal Blocks
• Feynman Rules for Scalar Conformal Blocks
• Dimensional reduction of higher-point conformal blocks
• Missing Corner in the Sky: Massless Three-Point Celestial Amplitudes
• Conformal Blocks in Two and Four Dimensions from Oscillator Representations
• Spinors and conformal correlators
• Superconformal Block from Holographic Geometry
• Crossing Symmetry for Long Multiplets in 4D $\mathcal{N}=1$ SCFTs