Distributions in CFT I. Cross-Ratio Space

Abstract

We show that the four-point functions in conformal field theory are defined as distributions on the boundary of the region of convergence of the conformal block expansion. The conformal block expansion converges in the sense of distributions on this boundary, i.e. it can be integrated term by term against appropriate test functions. This can be interpreted as a giving a new class of functionals that satisfy the swapping property when applied to the crossing equation, and we comment on the relation of our construction to other types of functionals. Our language is useful in all considerations involving the boundary of the region of convergence, e.g. for deriving the dispersion relations. We establish our results by elementary methods, relying only on crossing symmetry and the standard convergence properties of the conformal block expansion. This is the first in a series of papers on distributional properties of correlation functions in conformal field theory.

arXiv comment24 pages + appendices

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• The EFT Bootstrap at Finite $M_{PL}$
• Mellin amplitudes for 1$d$ CFT
• Ising model close to $d=2$
• Conformal Regge Theory at Finite Boost
• Extremal Chaos
• Conformal partial waves in momentum space
• Mixed Correlator Dispersive CFT Sum Rules
• Snowmass White Paper: The Quest to Define QFT
• Positivity, low twist dominance and CSDR for CFTs
• Isometry invariance of exact correlation functions in various charts of Minkowski and de Sitter spaces
• Classification of Convergent OPE Channels for Lorentzian CFT Four-Point Functions
• Theorems for the Lightcone Bootstrap
• Bound on the central charge of CFTs in large dimension
• From conformal correlators to analytic S-matrices: CFT${}_1$/QFT${}_2$