A Crossing-Symmetric OPE Inversion Formula

Abstract

We derive a Lorentzian OPE inversion formula for the principal series of $sl(2,\mathbb{R})$. Unlike the standard Lorentzian inversion formula in higher dimensions, the formula described here only applies to fully crossing-symmetric four-point functions and makes crossing symmetry manifest. In particular, inverting a single conformal block in the crossed channel returns the coefficient function of the crossing-symmetric sum of Witten exchange diagrams in AdS, including the direct-channel exchange. The inversion kernel exhibits poles at the double-trace scaling dimensions, whose contributions must cancel out in a generic solution to crossing. In this way the inversion formula leads to a derivation of the Polyakov bootstrap for $sl(2,\mathbb{R})$. The residues of the inversion kernel at the double-trace dimensions give rise to analytic bootstrap functionals discussed in recent literature, thus providing an alternative explanation for their existence. We also use the formula to give a general proof that the coefficient function of the principal series is meromorphic in the entire complex plane with poles only at the expected locations.

Mathematica notebook included, v2: references added

References in corpus (37)

• A bound on chaos
• Gapless Spin-Fluid Ground State in a Random Quantum Heisenberg Magnet
• Bounding scalar operator dimensions in 4D CFT
• The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
• The Analytic Bootstrap and AdS Superhorizon Locality
• Analyticity in Spin in Conformal Theories
• Two-dimensional conformal field theory and the butterfly effect
• Convexity and Liberation at Large Spin
• Bootstrapping Conformal Field Theories with the Extremal Functional Method
• Light-ray operators in conformal field theory
• Large Spin Perturbation Theory
• Loops in AdS from Conformal Field Theory
• Bootstrapping the 3d Ising twist defect
• A Mellin space approach to the conformal bootstrap
• A spacetime derivation of the Lorentzian OPE inversion formula
• Conformal Bootstrap in Mellin Space
• Gravitational S-matrix from CFT dispersion relations
• Loop Corrections to Supergravity on $Ad{S}_5\times S^5$
• Bounding the Space of Holographic CFTs with Chaos
• Half-BPS Wilson loop and AdS${}_2$/CFT${}_1$
• Analytic Bounds and Emergence of ${\text{AdS}}_2$ Physics from the Conformal Bootstrap
• All Tree-Level Correlators in AdS${}_5\times$S${}_5$ Supergravity: Hidden Ten-Dimensional Conformal Symmetry
• $d$-dimensional SYK, AdS Loops, and $6j$ Symbols
• Bootstrapping the half-BPS line defect
• The Analytic Functional Bootstrap I: 1D CFTs and 2D S-Matrices
• The Analytic Functional Bootstrap II: Natural Bases for the Crossing Equation
• A tauberian theorem for the conformal bootstrap
• Genus-One String Amplitudes from Conformal Field Theory
• Crossing Symmetry in Alpha Space
• Taming the $\epsilon$-expansion with Large Spin Perturbation Theory
• On the Polyakov-Mellin bootstrap
• Mellin space bootstrap for global symmetry
• Recursion Relations in Witten Diagrams and Conformal Partial Waves
• Simplifying large spin bootstrap in Mellin space
• Anomalous Dimensions from Crossing Kernels
• Cut-touching linear functionals in the conformal bootstrap
• Mellin Amplitudes for Fermionic Conformal Correlators

Cited by in corpus (29)

• Dispersive CFT Sum Rules
• AdS Bulk Locality from Sharp CFT Bounds
• Sphere Packing and Quantum Gravity
• Nonperturbative Mellin Amplitudes: Existence, Properties, Applications
• An Analytic Approach to BCFT${}_d$
• Crossing Symmetric Dispersion Relations for Mellin Amplitudes
• Bootstrapping the half-BPS line defect CFT in $\mathcal{N}=4$ SYM at strong coupling
• Harmonic analysis of 2d CFT partition functions
• Analytic Functional Bootstrap for CFTs in $d>1$
• Crossing symmetry, transcendentality and the Regge behaviour of 1d CFTs
• Bootstrapping line defects in $\mathcal{N}=2$ theories
• 2d Galilean Field Theories with Anisotropic Scaling
• On Galilean conformal bootstrap
• Notes on Resonances and Unitarity from Celestial Amplitudes
• Mellin amplitudes for 1$d$ CFT
• Dispersive Sum Rules in AdS${}_2$
• Higher-Dimensional Symmetry of AdS${}_2\times$S${}^2$ Correlators
• Conformal partial waves in momentum space
• The Unique Polyakov Blocks
• Mixed Correlator Dispersive CFT Sum Rules
• Celestial Mellin Amplitude
• Positivity, low twist dominance and CSDR for CFTs
• Notes on $n$-point Witten diagrams in AdS${}_2$
• Conformal Bootstrap Deformations
• The Sky Remembers everything: Celestial amplitude, Shadow and OPE in quadratic EFT of gravity
• Charging Up the Functional Bootstrap
• Bootstrability in Defect CFT: Integrated Correlators and Sharper Bounds
• 1d Conformal Field Theory and Dispersion Relations
• Fine Spectrum from Crude Analytic Bootstrap