Dispersion relations and exact bounds on CFT correlators

Abstract

We derive new crossing-symmetric dispersion formulae for CFT correlators restricted to the line. The formulae are equivalent to the sum rules implied by what we call master functionals, which are analytic extremal functionals which act on the crossing equation. The dispersion relations provide an equivalent formulation of the constraints of the Polyakov bootstrap and hence of crossing symmetry on the line. The built in positivity properties imply simple and exact lower and upper bounds on the values of general CFT correlators on the Euclidean section, which are saturated by generalized free fields. Besides bounds on correlators, we apply this technology to determine new universal constraints on the Regge limit of arbitrary CFTs and obtain very simple and accurate representations of the 3d Ising spin correlator.

arXiv comment46 pages + appendices

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• Crossing Symmetric Dispersion Relations for Mellin Amplitudes
• Quantum field theory and the Bieberbach conjecture
• Positivity and Geometric Function Theory Constraints on Pion Scattering
• QFT, EFT and GFT
• Towards Bootstrapping RG flows: Sine-Gordon in AdS
• Cosmological Dark Matter from a Bulk Black Hole
• Crossing antisymmetric Polyakov blocks + Dispersion relation
• Dispersive Sum Rules in AdS${}_2$
• Bounding 3d CFT correlators
• Locality and Analyticity of the Crossing Symmetric Dispersion Relation
• Conformal Bootstrap Deformations
• Positivity, low twist dominance and CSDR for CFTs
• Demystifying integrable QFTs in AdS: No-go theorems for higher-spin charges
• From conformal correlators to analytic S-matrices: CFT${}_1$/QFT${}_2$