Loops in AdS from Conformal Field Theory

Abstract

We propose and demonstrate a new use for conformal field theory (CFT) crossing equations in the context of AdS/CFT: the computation of loop amplitudes in AdS, dual to non-planar correlators in holographic CFTs. Loops in AdS are largely unexplored, mostly due to technical difficulties in direct calculations. We revisit this problem, and the dual $1/N$ expansion of CFTs, in two independent ways. The first is to show how to explicitly solve the crossing equations to the first subleading order in $1/N^2$, given a leading order solution. This is done as a systematic expansion in inverse powers of the spin, to all orders. These expansions can be resummed, leading to the CFT data for finite values of the spin. Our second approach involves Mellin space. We show how the polar part of the four-point, loop-level Mellin amplitudes can be fully reconstructed from the leading-order data. The anomalous dimensions computed with both methods agree. In the case of ${\varphi }^4$ theory in AdS, our crossing solution reproduces a previous computation of the one-loop bubble diagram. We can go further, deriving part of the four-point function in ${\varphi }^3+{\varphi }^4$ theory in AdS which had never been computed. In the process, we show how to analytically derive anomalous dimensions from Mellin amplitudes with an infinite series of poles, and discuss applications to more complicated cases such as the $\mathcal{N}=4$ super-Yang-Mills theory.

47+12 pages. v3: corrected typos, added reference, discussed subtlety of non-1PI diagrams

References in corpus (52)

• Gauge Theory Correlators from Non-Critical String Theory
• The Large N Limit of Superconformal Field Theories and Supergravity
• Holographic Derivation of Entanglement Entropy from AdS/CFT
• A bound on chaos
• One-Loop n-Point Gauge Theory Amplitudes, Unitarity and Collinear Limits
• Harmonic Polylogarithms
• Fusing Gauge Theory Tree Amplitudes Into Loop Amplitudes
• Quantum corrections to holographic entanglement entropy
• Holography from Conformal Field Theory
• The Amplituhedron
• Writing CFT correlation functions as AdS scattering amplitudes
• Causality Constraints on Corrections to the Graviton Three-Point Coupling
• The Analytic Bootstrap and AdS Superhorizon Locality
• Convexity and Liberation at Large Spin
• Universality of Long-Distance AdS Physics from the CFT Bootstrap
• Conformal Regge theory
• Graviton exchange and complete 4-point functions in the AdS/CFT correspondence
• A Natural Language for AdS/CFT Correlators
• Emergent Spacetime and Holographic CFTs
• Comments on operators with large spin
• One-loop Partition Functions of 3D Gravity
• Spinning AdS Propagators
• Causality Constraints in Conformal Field Theory
• Unitarity and the Holographic S-Matrix
• Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks
• Mellin amplitudes for $Ad{S}_5\times S^5$
• Towards Feynman rules for Mellin amplitudes in AdS/CFT
• A Mellin space approach to the conformal bootstrap
• AdS/CFT 4-point functions: How to succeed at z-integrals without really trying
• Analyticity and the Holographic S-Matrix
• Conformal Bootstrap in Mellin Space
• One Loop Tests of Higher Spin AdS/CFT
• Bounding the Space of Holographic CFTs with Chaos
• R^4 terms in 11 dimensions and conformal anomaly of (2,0) theory
• A Proof of the Conformal Collider Bounds
• Higher spins in AdS_5 at one loop: vacuum energy, boundary conformal anomalies and AdS/CFT
• Correlation Functions and Massive Kaluza-Klein Modes in the AdS/CFT Correspondence
• The Operator Product Expansion of N=4 SYM and the 4-point Functions of Supergravity
• AdS Field Theory from Conformal Field Theory
• The Heat Kernel on AdS_3 and its Applications
• General Scalar Exchange in AdS_d+1
• Comments on 4-point functions in the CFT/AdS correspondence
• 5 loops in 24/5 dimensions
• Conformal Bootstrap With Slightly Broken Higher Spin Symmetry
• More Holography from Conformal Field Theory
• On Feynman rules for Mellin amplitudes in AdS/CFT
• A New Spin on Causality Constraints
• Non-Abelian Binding Energies from the Lightcone Bootstrap
• 1/N^2 corrections to the holographic Weyl anomaly
• Order 1/N^2 test of the Maldacena conjecture: Cancellation of the one-loop Weyl anomaly
• Exploring Perturbative Conformal Field Theory in Mellin space
• Mellin transforming the minimal model CFTs: AdS/CFT at strong curvature

Cited by in corpus (131)

• The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
• Analyticity in Spin in Conformal Theories
• Large Spin Perturbation Theory
• Bootstrapping Inflationary Correlators in Mellin Space
• How to Succeed at Holographic Correlators Without Really Trying
• Operator bases, $S$-matrices, and their partition functions
• Gravitational S-matrix from CFT dispersion relations
• Loop Corrections to Supergravity on $Ad{S}_5\times S^5$
• $\mathcal{N}=4$ Super-Yang-Mills Correlators at Strong Coupling from String Theory and Localization
• Cosmological Cutting Rules
• All Tree-Level Correlators in AdS${}_5\times$S${}_5$ Supergravity: Hidden Ten-Dimensional Conformal Symmetry
• Quantum Gravity from Conformal Field Theory
• $d$-dimensional SYK, AdS Loops, and $6j$ Symbols
• $20^{\prime }$ Five-Point Function from $Ad{S}_5\times S^5$ Supergravity
• From dS to AdS and back
• The Conformal Bootstrap at Finite Temperature
• Cutting Cosmological Correlators
• The Analytic Functional Bootstrap II: Natural Bases for the Crossing Equation
• From Locality and Unitarity to Cosmological Correlators
• Where is String Theory?
• A Study of Quantum Field Theories in AdS at Finite Coupling
• Unmixing 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
• The Inflationary Wavefunction from Analyticity and Factorization
• On duality of color and kinematics in (A)dS momentum space
• Loop corrections for Kaluza-Klein AdS amplitudes
• 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
• Holographic Reconstruction of AdS Exchanges from Crossing Symmetry
• Conformal Bootstrap in the Regge Limit
• 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
• Spinning AdS Loop Diagrams: Two Point Functions
• Solving M-theory with the Conformal Bootstrap
• Nonperturbative Mellin Amplitudes: Existence, Properties, Applications
• The Mellin Formalism for Boundary CFT${}_d$
• The M-Theory S-Matrix From ABJM: Beyond 11D Supergravity
• New Modular Invariants in $\mathcal{N}=4$ Super-Yang-Mills Theory
• One-loop amplitudes in $Ad{S}_5\times S^5$ supergravity from $\mathcal{N}=4$ SYM at strong coupling
• Unitarity Methods in AdS/CFT
• CFT Unitarity and the AdS Cutkosky Rules
• On Superconformal Four-Point Mellin Amplitudes in Dimension $d>2$
• Double Copy Relation for AdS
• Recursion Relations in Witten Diagrams and Conformal Partial Waves
• Holographic Four-Point Functions in the (2, 0) Theory
• Bootstrapping Large Graviton non-Gaussianities
• Quantum ${\varphi }^4$ Theory in AdS${}_4$ and its CFT Dual
• Bootstrapping the half-BPS line defect CFT in $\mathcal{N}=4$ SYM at strong coupling
• From Amplitudes to Contact Cosmological Correlators
• Loops in Anti-de Sitter Space
• Harnessing S-Duality in $\mathcal{N}=4$ SYM & Supergravity as $SL(2,\mathbb{Z})$-Averaged Strings
• Veneziano Amplitude of Vasiliev Theory
• M-Theory Reconstruction from (2,0) CFT and the Chiral Algebra Conjecture
• Holographic dual of the five-point conformal block
• Simplifying large spin bootstrap in Mellin space
• A Crossing-Symmetric OPE Inversion Formula
• Loops in AdS: From the Spectral Representation to Position Space
• Growing Extra Dimensions in AdS/CFT
• Crossing symmetry, transcendentality and the Regge behaviour of 1d CFTs
• Scattering Equations in AdS: Scalar Correlators in Arbitrary Dimensions
• One-loop Gluon Amplitudes in AdS
• AdS Weight Shifting Operators
• Geodesic bulk diagrams on the Bruhat-Tits tree
• Black Holes and Conformal Regge Bootstrap
• From bulk loops to boundary large-N expansion
• All Holographic Four-Point Functions in All Maximally Supersymmetric CFTs
• On spinning loop amplitudes in Anti-de Sitter space
• Maximal $U(1{)}_Y$-violating $n$-point correlators in $\mathcal{N}=4$ super-Yang-Mills theory
• ABJM at Strong Coupling from M-theory, Localization, and Lorentzian Inversion
• On non-supersymmetric conformal manifolds: field theory and holography
• Modular invariant holographic correlators for $\mathcal{N}=4$ SYM with general gauge group
• Leading Order Anomalous Dimensions at the Wilson-Fisher Fixed Point from CFT
• One-loop string corrections to AdS amplitudes from CFT
• Dispersion Formulas in QFTs, CFTs, and Holography
• Bootstrapping the $(A_1,A_2)$ Argyres-Douglas theory
• 6d (2,0) and M-theory at 1-loop
• On Mellin Amplitudes in SCFTs with Eight Supercharges
• Higher Spin ANEC and the Space of CFTs
• AdS${}_4$/CFT${}_3$ for Unprotected Operators
• All loop structures in Supergravity Amplitudes on $Ad{S}_5\times S^5$ from CFT
• Spinor-Helicity Formalism for Massless Fields in AdS${}_4$ II: Potentials
• Integrated correlators from integrability: Maldacena-Wilson line in $\mathcal{N}=4$ SYM
• Opacity from Loops in AdS
• Model-Dependence of Minimal-Twist OPEs in $d>2$ Holographic CFTs
• Analytical evaluation of AdS${}_4$ Witten diagrams as flat space multi-loop Feynman integrals
• Mellin Amplitudes for Fermionic Conformal Correlators
• Swampland Conditions for Higher Derivative Couplings from CFT
• Loops in AdS: From the Spectral Representation to Position Space II
• On Holography in General Background and the Boundary Effective Action from AdS to dS
• Opacity and Effective Field Theory in Anti-de Sitter Backgrounds
• Towards All Loop Supergravity Amplitudes on $Ad{S}_5\times S^5$
• Polyakov-Mellin Bootstrap for AdS loops
• Spinor-Helicity Formalism for Massless Fields in AdS${}_4$ III: Contact Four-Point Amplitudes
• Effective Field Theory in AdS: Continuum Regime, Soft Bombs, and IR Emergence
• On one loop corrections in higher spin gravity
• More on holographic correlators: Twisted and dimensionally reduced structures
• AdS super gluon scattering up to two loops: A position space approach
• AdS/CFT Unitarity at Higher Loops: High-Energy String Scattering
• Bootstrapping string dynamics in the 6d $\mathcal{N}=(2,0)$ theories
• ABJ Correlators with Weakly Broken Higher Spin Symmetry
• The perturbative CFT optical theorem and high-energy string scattering in AdS at one loop
• Bulk renormalization and the AdS/CFT correspondence
• A Scattering Amplitude for Massive Particles in AdS
• Geodesic description of Heavy-Light Virasoro blocks
• Pure anti-de Sitter supergravity and the conformal bootstrap
• Exploring Confinement in Anti-de Sitter Space
• Bootstrapping mixed correlators in $\mathcal{N}=4$ Super Yang-Mills
• Celestial amplitude for 2d theory
• Holographic S-fold theories at one loop
• Bootstrapping Witten diagrams via differential representation in Mellin space
• The Inversion Formula and 6j Symbol for 3d Fermions
• Recursion Relations in $p$-adic Mellin Space
• Three point functions in higher spin AdS_3 holography with 1/N corrections
• Scattering bound states in AdS
• Renormalization Group Flows, the $a$-Theorem and Conformal Bootstrap
• The Analytic Bootstrap for Large $N$ Chern-Simons Vector Models
• All Next-to-Next-to-Extremal One-Loop Correlators of AdS Supergluons and Supergravitons
• Mellin amplitudes for $Ad{S}_3\times S^3$
• c-Recursion for multi-point superconformal blocks. NS sector
• Bootstrability in Defect CFT: Integrated Correlators and Sharper Bounds
• Recursive construction of the operator product expansion in curved space
• Correlation functions at the bulk point singularity from the gravitational eikonal S-matrix
• Information Geometry and Holographic Correlators
• Conformal bootstrap in momentum space at finite volume
• Rebooting quarter-BPS operators in $\mathcal{N}=4$ Super Yang-Mills
• Composite AdS geodesics for CFT correlators and timelike entanglement entropy
• Constructing CFTs from AdS flows