Analytic Functional Bootstrap for CFTs in $d>1$

Abstract

We introduce analytic functionals which act on the crossing equation for CFTs in arbitrary spacetime dimension. The functionals fully probe the constraints of crossing symmetry on the first sheet, and are in particular sensitive to the OPE, (double) lightcone and Regge limits. Compatibility with the crossing equation imposes constraints on the functional kernels which we study in detail. We then introduce two simple classes of functionals. The first class has a simple action on generalized free fields and their deformations and can be used to bootstrap AdS contact interactions in general dimension. The second class is obtained by tensoring holomorphic and antiholomorphic copies of $d=1$ functionals which have been considered recently. They are dual to simple solutions to crossing in $d=2$ which include the energy correlator of the Ising model. We show how these functionals lead to optimal bounds on the OPE density of $d=2$ CFTs and argue that they provide an equivalent rewriting of the $d=2$ crossing equation which is better suited for numeric computations than current approaches.

40 pages + appendices V3. Minor typos fixed

References in corpus (10)

• Holography from Conformal Field Theory
• Writing CFT correlation functions as AdS scattering amplitudes
• The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT
• Conformal Bootstrap in Mellin Space
• A tauberian theorem for the conformal bootstrap
• Sphere Packing and Quantum Gravity
• Modular Invariance, Tauberian Theorems, and Microcanonical Entropy
• An Analytic Approach to BCFT${}_d$
• Tauberian-Cardy formula with spin
• A Crossing-Symmetric OPE Inversion Formula

Cited by in corpus (29)

• Sharp Boundaries for the Swampland
• Free partition functions and an averaged holographic duality
• Dispersive CFT Sum Rules
• AdS Bulk Locality from Sharp CFT Bounds
• Selected Topics in Analytic Conformal Bootstrap: A Guided Journey
• 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
• Distributions in CFT I. Cross-Ratio Space
• Universality in asymptotic bounds and its saturation in $2$D CFT
• A Scattering Amplitude in Conformal Field Theory
• On Galilean conformal bootstrap
• Crossing antisymmetric Polyakov blocks + Dispersion relation
• The Shadow Formalism of Galilean CFT${}_2$
• Dispersive Sum Rules in AdS${}_2$
• Lightcone expansions of conformal blocks in closed form
• Bounding 3d CFT correlators
• Conformal partial waves in momentum space
• The Unique Polyakov Blocks
• Mixed Correlator Dispersive CFT Sum Rules
• Conformal Bootstrap Deformations
• Classification of Convergent OPE Channels for Lorentzian CFT Four-Point Functions
• Charging Up the Functional Bootstrap
• Bound on the central charge of CFTs in large dimension
• Large $N$ analytical functional bootstrap I: 1D CFTs and total positivity
• How to Succeed at Witten Diagram Recursions without Really Trying
• Bootstrapping Boundaries and Branes
• Three Dimensional Pure Gravity and Generalized Hecke Operators