Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks

Abstract

We develop a new method for decomposing Witten diagrams into conformal blocks. The steps involved are elementary, requiring no explicit integration, and operate directly in position space. Central to this construction is an appealingly simple answer to the question: what object in AdS computes a conformal block? The answer is a "geodesic Witten diagram," which is essentially an ordinary exchange Witten diagram, except that the cubic vertices are not integrated over all of AdS, but only over bulk geodesics connecting the boundary operators. In particular, we consider the case of four-point functions of scalar operators, and show how to easily reproduce existing results for the relevant conformal blocks in arbitrary dimension.

54 pages + refs

References in corpus (13)

• Holography from Conformal Field Theory
• Writing CFT correlation functions as AdS scattering amplitudes
• Universal Spectrum of 2d Conformal Field Theory in the Large c Limit
• Virasoro Conformal Blocks and Thermality from Classical Background Fields
• Spinning AdS Propagators
• Holographic Entanglement Entropy from 2d CFT: Heavy States and Local Quenches
• Eikonal Approximation in AdS/CFT: Conformal Partial Waves and Finite N Four-Point Functions
• Universal anomalous dimensions at large spin and large twist
• Analytic bootstrap at large spin
• Four point function of $\mathcal{N}=4$ stress-tensor multiplet at strong coupling
• BCFW for Witten Diagrams
• Classical conformal blocks via AdS/CFT correspondence
• Conjectures for Large N Superconformal N=4 Chiral Primary Four Point Functions

Cited by in corpus (164)

• The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
• Causality Constraints in Conformal Field Theory
• A Stereoscopic Look into the Bulk
• Loops in AdS from Conformal Field Theory
• A Mellin space approach to the conformal bootstrap
• Conformal Bootstrap in Mellin Space
• Entanglement Wedge Cross Section from the Dual Density Matrix
• Bounding the Space of Holographic CFTs with Chaos
• All point correlation functions in SYK
• Weight Shifting Operators and Conformal Blocks
• $d$-dimensional SYK, AdS Loops, and $6j$ Symbols
• Entanglement, Holography and Causal Diamonds
• Einstein gravity 3-point functions from conformal field theory
• The Conformal Bootstrap at Finite Temperature
• The Analytic Functional Bootstrap I: 1D CFTs and 2D S-Matrices
• Quantum Regge Trajectories and the Virasoro Analytic Bootstrap
• Multipoint Conformal Blocks in the Comb Channel
• Spinning Witten Diagrams
• Bulk Phase Shift, CFT Regge Limit and Einstein Gravity
• Conformal Basis, Optical Theorem, and the Bulk Point Singularity
• Holographic Reconstruction of AdS Exchanges from Crossing Symmetry
• Hawking from Catalan
• Landau diagrams in AdS and S-matrices from conformal correlators
• Spinning AdS Loop Diagrams: Two Point Functions
• The Mellin Formalism for Boundary CFT${}_d$
• Higher-point conformal blocks and entanglement entropy in heavy states
• Momentum space approach to crossing symmetric CFT correlators
• An Analytic Approach to BCFT${}_d$
• The BMS Bootstrap
• Seed Conformal Blocks in 4D CFT
• Unitarity Methods in AdS/CFT
• Exact Virasoro Blocks from Wilson Lines and Background-Independent Operators
• AdS Maps and Diagrams of Bi-local Holography
• Recursion Relations in Witten Diagrams and Conformal Partial Waves
• From global to heavy-light: 5-point conformal blocks
• CFT in AdS and boundary RG flows
• Universal Lowest-Twist in CFTs from Holography
• A CFT Perspective on Gravitational Dressing and Bulk Locality
• Charged structure constants from modularity
• Holographic Entanglement and Poincare blocks in three dimensional flat space
• Celestial Amplitudes as AdS-Witten Diagrams
• Local bulk physics from intersecting modular Hamiltonians
• An étude of momentum space scalar amplitudes in AdS
• Simplifying large spin bootstrap in Mellin space
• Holographic dual of the five-point conformal block
• A Conformal Collider for Holographic CFTs
• New relation for AdS amplitudes
• Black Holes, Heavy States, Phase Shift and Anomalous Dimensions
• Quantum vs. classical information: operator negativity as a probe of scrambling
• A Bound on Massive Higher Spin Particles
• Correlators at large c without information loss
• Anatomy of Geodesic Witten Diagrams
• Witten Diagrams for Torus Conformal Blocks
• Holographic 4-point correlators with heavy states
• Loops in AdS: From the Spectral Representation to Position Space
• Towards Entanglement of Purification for Conformal Field Theories
• Virasoro vacuum block at next-to-leading order in the heavy-light limit
• Holographic Conformal Partial Waves as Gravitational Open Wilson Networks
• Holographic conformal blocks from interacting Wilson lines
• Lattice Setup for Quantum Field Theory in AdS${}_2$
• Bootstrapping line defects with $O(2)$ global symmetry
• Scattering Equations in AdS: Scalar Correlators in Arbitrary Dimensions
• Geodesic Diagrams, Gravitational Interactions & OPE Structures
• Gaudin Models and Multipoint Conformal Blocks: General Theory
• Towards the higher point holographic momentum space amplitudes II: Gravitons
• Universal Asymptotics for High Energy CFT Data
• AdS Weight Shifting Operators
• Higher-Point Conformal Blocks in the Comb Channel
• Dynamical Constraints on RG Flows and Cosmology
• Monodromy Defects from Hyperbolic Space
• From Conformal Blocks to Path Integrals in the Vaidya Geometry
• Geodesic bulk diagrams on the Bruhat-Tits tree
• Black Holes and Conformal Regge Bootstrap
• Scalar quasinormal modes of Kerr-AdS${}_5$
• On spinning loop amplitudes in Anti-de Sitter space
• Holographic OPE Coefficients from AdS Black Holes with Matters
• Thermal Conformal Blocks
• Interactions in Higher-Spin Gravity: a Holographic Perspective
• A multipoint conformal block chain in $d$ dimensions
• Helicity basis for three-dimensional conformal field theory
• Aspects of CFTs on Real Projective Space
• Confluent conformal blocks and the Teukolsky master equation
• Holographic Description of 2D Conformal Block in Semi-classical Limit
• Dispersion Formulas in QFTs, CFTs, and Holography
• Semi-classical BMS${}_3$ blocks and flat holography
• Spinning Geodesic Witten Diagrams
• Seven-Point Conformal Blocks in the Extended Snowflake Channel and Beyond
• Double-Trace Deformations of Conformal Correlations
• Multi-centered AdS${}_3$ solutions from Virasoro conformal blocks
• Holographic Entropy Cone with Time Dependence in Two Dimensions
• Thermalization in Large-N CFTs
• Operator mixing in deformed D1D5 CFT and the OPE on the cover
• Six-Point Conformal Blocks in the Snowflake Channel
• Perturbative classical conformal blocks as Steiner trees on the hyperbolic disk
• New Methods for Conformal Correlation Functions
• Opacity from Loops in AdS
• Equivalent Equations of Motion for Gravity and Entropy
• Renormalization of gravitational Wilson lines
• Towards Feynman rules for conformal blocks
• Chaos in CFT dual to rotating BTZ
• Locality, bulk equations of motion and the conformal bootstrap
• Swampland Conditions for Higher Derivative Couplings from CFT
• Airy Function and 4d Quantum Gravity
• Loops in AdS: From the Spectral Representation to Position Space II
• Conformal Floquet dynamics with a continuous drive protocol
• Bootstrapping closed string field theory
• Large $c$ Virasoro Blocks from Monodromy Method beyond Known Limits
• On conformal correlators and blocks with spinors in general dimensions
• Fermions in Geodesic Witten Diagrams
• Towards a Bootstrap approach to higher orders of epsilon expansion
• New Properties of Large-$c$ Conformal Blocks from Recursion Relation
• Scalar Blocks as Gravitational Wilson Networks
• A note on the Virasoro blocks at order $1/c$
• Hyperbolic Lattice for Scalar Field Theory in AdS${}_3$
• Emergent gravity from Eguchi-Kawai reduction
• Kinematic space for conical defects
• Bands and gaps in Nekrasov partition function
• Boundary Kinematic Space
• Efficient Rules for All Conformal Blocks
• Structure Constants from Modularity in Warped CFT
• Geodesic Witten diagrams with anti-symmetric exchange
• The AdS${}_3$ Propagator and the Fate of Locality
• Exploring Perturbative Conformal Field Theory in Mellin space
• Dimensional reduction of higher-point conformal blocks
• Correlation functions in $\mathcal{N}=2$ Supersymmetric vector matter Chern-Simons theory
• Multi-centered higher spin solutions from $W_N$ conformal blocks
• More on Wilson toroidal networks and torus blocks
• On Conformal Block, Crossing Kernel and Multi-variable Hypergeometric Functions
• Bounding 3d CFT correlators
• Regge conformal blocks from the Rindler-AdS black hole and the pole-skipping phenomena
• Tensor network state correspondence and holography
• Does boundary quantum mechanics imply quantum mechanics in the bulk?
• Toward the Pole
• Bootstrapping Witten diagrams via differential representation in Mellin space
• Higher-spin gravity's "string": new gauge and proof of holographic duality for the linearized Didenko-Vasiliev solution
• Light-state Dominance from the Conformal Bootstrap
• On skew tau-functions in higher spin theory
• Holographic relations for OPE blocks in excited states
• Holographic equivalence between the first law of entanglement entropy and the linearized gravitational equations
• Holographic Reflected Entropy and Islands in Interface CFTs
• A geodesic Witten diagram description of holographic entanglement entropy and its quantum corrections
• Boundary Holographic Witten Diagrams
• Anomalous Dimensions from Thermal AdS Partition Functions
• Renormalization Group Flows, the $a$-Theorem and Conformal Bootstrap
• The Probe of Curvature in the Lorentzian AdS${}_2$/CFT${}_1$ Correspondence
• Berry Curvature and Riemann Curvature in Kinematic Space with Spherical Entangling Surface
• Bulk locality and gauge invariance for boundary-bilocal cubic correlators in higher-spin gravity
• More on Heavy-Light Bootstrap up to Double-Stress-Tensor
• c-Recursion for multi-point superconformal blocks. NS sector
• The gravity dual of Lorentzian OPE blocks
• Remarks on holographic models of the Kerr-AdS${}_5$ geometry
• From conformal correlators to analytic S-matrices: CFT${}_1$/QFT${}_2$
• Spinning Particle Geometries in AdS${}_3$/CFT${}_2$
• More on Boundary Holographic Witten Diagrams
• Holographic Complexity from the Crofton's Formula in Lorentzian AdS${}_3$
• Entanglement Entropy with a Time-dependent Hamiltonian
• Superconformal Block from Holographic Geometry
• Holographic BCFT Spectra from Brane Mergers
• Correlation function with the insertion of zero modes of modular Hamiltonians
• Toward Holography on Biregular Trees
• Recursive structure of the Gauß hypergeometric function and boundary/crosscap conformal block
• On scalar propagators of three-dimensional higher-spin black holes
• Moments and saddles of heavy CFT correlators
• Composite AdS geodesics for CFT correlators and timelike entanglement entropy