How to Succeed at Holographic Correlators Without Really Trying

Abstract

We give a detailed account of the methods introduced in [1] to calculate holographic four-point correlators in IIB supergravity on $Ad{S}_5\times S^5$. Our approach relies entirely on general consistency conditions and maximal supersymmetry. We discuss two related methods, one in position space and the other in Mellin space. The position space method is based on the observation that the holographic four-point correlators of one-half BPS single-trace operators can be written as finite sums of contact Witten diagrams. We demonstrate in several examples that imposing the superconformal Ward identity is sufficient to fix the parameters of this ansatz uniquely, avoiding the need for a detailed knowledge of the supergravity effective action. The Mellin space approach is an "on-shell method" inspired by the close analogy between holographic correlators and flat space scattering amplitudes. We conjecture a compact formula for the four-point correlators of one-half BPS single-trace operators of arbitrary weights. Our general formula has the expected analytic structure, obeys the superconformal Ward identity, satisfies the appropriate asymptotic conditions and reproduces all the previously calculated cases. We believe that these conditions determine it uniquely.

56 pages. A Mathematica notebook attached to the submission contains some explicit results both in position and in Mellin space; v2 published version

References in corpus (12)

• Writing CFT correlation functions as AdS scattering amplitudes
• Spinning AdS Propagators
• Mellin amplitudes for $Ad{S}_5\times S^5$
• Conformal Bootstrap in Mellin Space
• Loop Corrections to Supergravity on $Ad{S}_5\times S^5$
• Hexagonalization of Correlation Functions
• A non-renormalization theorem for chiral primary 3-point functions
• BCFW for Witten Diagrams
• Conjectures for Large N Superconformal N=4 Chiral Primary Four Point Functions
• A Note on (Non)-Locality in Holographic Higher Spin Theories
• Four-point correlators with higher weight superconformal primaries in the AdS/CFT Correspondence
• Exploring Perturbative Conformal Field Theory in Mellin space

Cited by in corpus (110)

• Gravitational S-matrix from CFT dispersion relations
• $\mathcal{N}=4$ Super-Yang-Mills Correlators at Strong Coupling from String Theory and Localization
• All Tree-Level Correlators in AdS${}_5\times$S${}_5$ Supergravity: Hidden Ten-Dimensional Conformal Symmetry
• $20^{\prime }$ Five-Point Function from $Ad{S}_5\times S^5$ Supergravity
• Selected Topics in Analytic Conformal Bootstrap: A Guided Journey
• Genus-One String Amplitudes from Conformal Field Theory
• Gluon Scattering in AdS from CFT
• Far Beyond the Planar Limit in Strongly-Coupled $\mathcal{N}=4$ SYM
• Bootstrapping the simplest correlator in planar N = 4 SYM at all loops
• The double-trace spectrum of N=4 SYM at strong coupling
• Genus-2 Holographic Correlator on $Ad{S}_5\times S^5$ from Localization
• Modular Invariance in Superstring Theory From $\mathcal{N}=4$ Super-Yang-Mills
• Handling Handles: Nonplanar Integrability in $\mathcal{N}=4$ Supersymmetric Yang-Mills Theory
• $\mathbf{Ad}{\mathbf{S}}_3\times {\mathbf{S}}^3$ Tree-Level Correlators: Hidden Six-Dimensional Conformal Symmetry
• On the Polyakov-Mellin bootstrap
• Nonperturbative Mellin Amplitudes: Existence, Properties, Applications
• New Modular Invariants in $\mathcal{N}=4$ Super-Yang-Mills Theory
• The M-Theory S-Matrix From ABJM: Beyond 11D Supergravity
• One-loop amplitudes in $Ad{S}_5\times S^5$ supergravity from $\mathcal{N}=4$ SYM at strong coupling
• Perturbative Four-Point Functions In Planar N=4 SYM From Hexagonalization
• CFT Unitarity and the AdS Cutkosky Rules
• On Superconformal Four-Point Mellin Amplitudes in Dimension $d>2$
• Double copy structure and the flat space limit of conformal correlators in even dimensions
• Double Copy Relation for AdS
• Recursion Relations in Witten Diagrams and Conformal Partial Waves
• A Note on (Non)-Locality in Holographic Higher Spin Theories
• Holographic Four-Point Functions in the (2, 0) Theory
• Cosmological Scattering Equations
• AdS${}_4$/CFT${}_3$ from Weak to Strong String Coupling
• Harnessing S-Duality in $\mathcal{N}=4$ SYM & Supergravity as $SL(2,\mathbb{Z})$-Averaged Strings
• All Tree-Level Correlators for M-theory on $Ad{S}_7\times S^4$
• M-Theory Reconstruction from (2,0) CFT and the Chiral Algebra Conjecture
• Defect CFT in the 6d (2,0) theory from M2 brane dynamics in AdS${}_7\times$S${}^4$
• On the Differential Representation and Color-Kinematics Duality of AdS Boundary Correlators
• Holographic correlators in AdS${}_3$ without Witten diagrams
• Holographic correlators in AdS${}_3$
• Loops in AdS: From the Spectral Representation to Position Space
• Single Particle Operators and their Correlators in Free $\mathcal{N}=4$ SYM
• The CFT${}_6$ origin of all tree-level 4-point correlators in AdS${}_3\times S^3$
• Growing Extra Dimensions in AdS/CFT
• AdS${}_3$ Holography at Dimension Two
• String corrections to AdS amplitudes and the double-trace spectrum of N=4 SYM
• The Cosmological Bootstrap: Spinning Correlators from Symmetries and Factorization
• One-loop Gluon Amplitudes in AdS
• Homotopy Properties and Lower-Order Vertices in Higher-Spin Equations
• Bootstrapping holographic defect correlators in $\mathcal{N}=4$ super Yang-Mills
• Super Gluon Five-Point Amplitudes in AdS Space
• Towards the Virasoro-Shapiro Amplitude in AdS5xS5
• Supercharged AdS${}_3$ Holography
• Black Holes and Conformal Regge Bootstrap
• All Holographic Four-Point Functions in All Maximally Supersymmetric CFTs
• From bulk loops to boundary large-N expansion
• Comments on the $S_N$ orbifold CFT in the large $N$-limit
• Maximal $U(1{)}_Y$-violating $n$-point correlators in $\mathcal{N}=4$ super-Yang-Mills theory
• The Virasoro-Shapiro amplitude in AdS${}_5\times$S${}^5$ and level splitting of 10d conformal symmetry
• ABJM at Strong Coupling from M-theory, Localization, and Lorentzian Inversion
• Modular invariant holographic correlators for $\mathcal{N}=4$ SYM with general gauge group
• Four-point functions of 1/2-BPS operators of any weights in the supergravity approximation
• One-loop string corrections to AdS amplitudes from CFT
• 6d (2,0) and M-theory at 1-loop
• The SAGEX Review on Scattering Amplitudes, Chapter 8: Half BPS correlators
• Localization vs holography in $4d$ $\mathcal{N}=2$ quiver theories
• All five-loop planar four-point functions of half-BPS operators in $\mathcal{N}=4$ SYM
• On Mellin Amplitudes in SCFTs with Eight Supercharges
• All loop structures in Supergravity Amplitudes on $Ad{S}_5\times S^5$ from CFT
• AdS${}_4$/CFT${}_3$ for Unprotected Operators
• Bootstrapping 4d $\mathcal{N}=2$ gauge theories: the case of SQCD
• Thermal Stress Tensor Correlators, OPE and Holography
• An alternative to diagrams for the critical O(N) model: dimensions and structure constants to order $1/N^2$
• AdS${}_3$ four-point functions from $\frac{1}{8}$-BPS states
• BCJ relations in ${AdS}_5\times S^3$ and the double-trace spectrum of super gluons
• The AdS Veneziano amplitude at small curvature
• Laplace-difference equation for integrated correlators of operators with general charges in $\mathcal{N}=4$ SYM
• Structure constants in $\mathcal{N}=2$ superconformal quiver theories at strong coupling and holography
• Loops in AdS: From the Spectral Representation to Position Space II
• Four-point functions of all-different-weight chiral primary operators in the supergravity approximation
• More on holographic correlators: Twisted and dimensionally reduced structures
• Crossing antisymmetric Polyakov blocks + Dispersion relation
• AdS super gluon scattering up to two loops: A position space approach
• On one loop corrections in higher spin gravity
• Two-particle spectrum of tensor multiplets coupled to $Ad{S}_3\times S^3$ gravity
• One-loop string corrections for AdS Kaluza-Klein amplitudes
• Mellin amplitudes for 1$d$ CFT
• Bootstrapping string dynamics in the 6d $\mathcal{N}=(2,0)$ theories
• ABJ Correlators with Weakly Broken Higher Spin Symmetry
• Higher-Dimensional Symmetry of AdS${}_2\times$S${}^2$ Correlators
• Spinning correlators in $\mathcal{N}=2$ SCFTs: Superspace and AdS amplitudes
• Notes on gravity multiplet correlators in $Ad{S}_3\times S^3$
• Yangian Symmetry in Holographic Correlators
• Pure anti-de Sitter supergravity and the conformal bootstrap
• Revisiting Extremal Couplings in AdS/CFT
• Bootstrapping Witten diagrams via differential representation in Mellin space
• Celestial Mellin Amplitude
• Holographic S-fold theories at one loop
• Recursion Relations in $p$-adic Mellin Space
• Notes on $n$-point Witten diagrams in AdS${}_2$
• One-loop string amplitudes in AdS${}_5\times$S${}^5$: Mellin space and sphere splitting
• Scattering bound states in AdS
• Bootstrapping string theory on AdS${}_5\times S^5$
• Four-point correlators in N=4 SYM from AdS${}_5$ bubbling geometries
• Bootstrapping AdS${}_2\times$ S${}^2$ hypermultiplets: hidden four-dimensional conformal symmetry
• Mellin amplitudes for $Ad{S}_3\times S^3$
• An Elliptic One-Loop Amplitude in Anti-de-Sitter Space
• BCJ Amplitude Relations for Anti-de Sitter Boundary Correlators in Embedding Space
• Conformal bootstrap in momentum space at finite volume
• The $\mathcal{W}$-algebra bootstrap of 6d $\mathcal{N}=(2,0)$ theories
• Rebooting quarter-BPS operators in $\mathcal{N}=4$ Super Yang-Mills
• Constructing CFTs from AdS flows
• Lepton flavor violating top quark FCNC processes at the $\mu$TRISTAN
• On-shell Correlators and Color-Kinematics Duality in Curved Symmetric Spacetimes