A Basis of Analytic Functionals for CFTs in General Dimension

Abstract

We develop an analytic approach to the four-point crossing equation in CFT, for general spacetime dimension. In a unitary CFT, the crossing equation (for, say, the s- and t-channel expansions) can be thought of as a vector equation in an infinite-dimensional space of complex analytic functions in two variables, which satisfy a boundedness condition in the u-channel Regge limit. We identify a useful basis for this space of functions, consisting of the set of s- and t-channel conformal blocks of double-twist operators in mean field theory. We describe two independent algorithms to construct the dual basis of linear functionals, and work out explicitly many examples. Our basis of functionals appears to be closely related to the CFT dispersion relation recently derived by Carmi and Caron-Huot.

47 pages, 2 figures; v2: references added, conventions fixed; v3: published version

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• AdS Bulk Locality from Sharp CFT Bounds
• All Tree-Level Correlators for M-theory on $Ad{S}_7\times S^4$
• Tauberian-Cardy formula with spin
• The Lorentzian inversion formula and the spectrum of the 3d O(2) CFT
• Beurling-Selberg Extremization and Modular Bootstrap at High Energies
• Dispersion relations and exact bounds on CFT correlators
• Crossing symmetry, transcendentality and the Regge behaviour of 1d CFTs
• Aspects of CFTs on Real Projective Space
• Dispersion Formulas in QFTs, CFTs, and Holography
• Crossing antisymmetric Polyakov blocks + Dispersion relation
• Bounding 3d CFT correlators
• Conformal partial waves in momentum space
• Lightcone expansions of conformal blocks in closed form
• Classification of Convergent OPE Channels for Lorentzian CFT Four-Point Functions
• Theorems for the Lightcone Bootstrap
• Charging Up the Functional Bootstrap
• The geometry of optimal functionals
• How to Succeed at Witten Diagram Recursions without Really Trying