Conformal correlators as simplex integrals in momentum space

Abstract

We find the general solution of the conformal Ward identities for scalar $n$-point functions in momentum space and in general dimension. The solution is given in terms of integrals over $(n-1)$-simplices in momentum space. The $n$ operators are inserted at the $n$ vertices of the simplex, and the momenta running between any two vertices of the simplex are the integration variables. The integrand involves an arbitrary function of momentum-space cross ratios constructed from the integration variables, while the external momenta enter only via momentum conservation at each vertex. Correlators where the function of cross ratios is a monomial exhibit a remarkable recursive structure where $n$-point functions are built in terms of $(n-1)$-point functions. To illustrate our discussion, we derive the simplex representation of $n$-point contact Witten diagrams in a holographic conformal field theory. This can be achieved through both a recursive method, as well as an approach based on the star-mesh transformation of electrical circuit theory. The resulting expression for the function of cross ratios involves $(n-2)$ integrations, which is an improvement (when $n>4$) relative to the Mellin representation that involves $n(n-3)/2$ integrations.

arXiv comment44 pages, 4 figures. v2: published version

References in corpus (84)

• Non-Gaussian features of primordial fluctuations in single field inflationary models
• The dS/CFT Correspondence
• The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
• Conformal Four Point Functions and the Operator Product Expansion
• Conformal Partial Waves and the Operator Product Expansion
• Writing CFT correlation functions as AdS scattering amplitudes
• On graviton non-Gaussianities during inflation
• Quantum Gravity In De Sitter Space
• Graviton exchange and complete 4-point functions in the AdS/CFT correspondence
• Implications of conformal invariance in momentum space
• Inflation and the dS/CFT Correspondence
• A Natural Language for AdS/CFT Correlators
• A Mellin space approach to the conformal bootstrap
• AdS/CFT 4-point functions: How to succeed at z-integrals without really trying
• Bootstrapping Inflationary Correlators in Mellin Space
• Scaled momentum distributions of charged particles in dijet photoproduction at HERA
• Conformal Partial Waves: Further Mathematical Results
• Conformal Bootstrap in Mellin Space
• New Recursion Relations and a Flat Space Limit for AdS/CFT Correlators
• The Cosmological Bootstrap: Weight-Shifting Operators and Scalar Seeds
• Local bulk S-matrix elements and CFT singularities
• A Mellin Space Approach to Cosmological Correlators
• CMB from CFT
• Scalar 3-point functions in CFT: renormalisation, beta functions and anomalies
• D-independent representation of Conformal Field Theories in D dimensions via transformation to auxiliary Dual Resonance Models. Scalar amplitudes
• Late-time Structure of the Bunch-Davies De Sitter Wavefunction
• Operator Product Expansion of Inflationary Correlators and Conformal Symmetry of de Sitter
• Solving the Conformal Constraints for Scalar Operators in Momentum Space and the Evaluation of Feynman's Master Integrals
• Holography for inflation using conformal perturbation theory
• From AdS to dS Exchanges: Spectral Representation, Mellin Amplitudes and Crossing
• Conformal Invariance, Dark Energy, and CMB Non-Gaussianity
• Conformal Invariance and the Four Point Scalar Correlator in Slow-Roll Inflation
• S-Matrices from AdS Spacetime
• TASI Lectures on the Conformal Bootstrap
• Double copy structure of CFT correlators
• Constraints from Conformal Symmetry on the Three Point Scalar Correlator in Inflation
• Renormalised 3-point functions of stress tensors and conserved currents in CFT
• Conformal $n$-point functions in momentum space
• Renormalised CFT 3-point functions of scalars, currents and stress tensors
• BCFW for Witten Diagrams
• Holographic predictions for cosmological 3-point functions
• Cosmological Polytopes and the Wavefunction of the Universe
• Four Point Functions of the Stress Tensor and Conserved Currents in AdS_4/CFT_3
• On the Polyakov-Mellin bootstrap
• Momentum space approach to crossing symmetric CFT correlators
• Lorentzian CFT 3-point functions in momentum space
• Conformal Anomalies and the Gravitational Effective Action: The $TJJ$ Correlator for a Dirac Fermion
• Double copy structure and the flat space limit of conformal correlators in even dimensions
• Multipoint correlators of conformal field theories: implications for quantum critical transport
• Evaluation of conformal integrals
• Momentum-space conformal blocks on the light cone
• Implications of N=1 Superconformal Symmetry for Chiral Fields
• TTT in CFT: Trace Identities and the Conformal Anomaly Effective Action
• An étude of momentum space scalar amplitudes in AdS
• On critical exponents without Feynman diagrams
• Momentum space conformal three-point functions of conserved currents and a general spinning operator
• New relation for AdS amplitudes
• On the four-point function of the stress-energy tensors in a CFT
• Exact Correlators from Conformal Ward Identities in Momentum Space and the Perturbative $TJJ$ Vertex
• Momentum space approach to crossing symmetric CFT correlators II: General spacetime dimension
• The Cosmological Bootstrap: Spinning Correlators from Symmetries and Factorization
• Towards the higher point holographic momentum space amplitudes II: Gravitons
• Conformal 3-point functions and the Lorentzian OPE in momentum space
• On Some Hypergeometric Solutions of the Conformal Ward Identities of Scalar 4-point Functions in Momentum Space
• On the Positive Geometry of Conformal Field Theory
• Graviton Vertices and the Mapping of Anomalous Correlators to Momentum Space for a General Conformal Field Theory
• On the power spectrum of inflationary cosmologies dual to a deformed CFT
• Cosmological Shapes of Higher-Spin Gravity
• Conserved current correlators of conformal field theories in 2+1 dimensions
• Multi-field inflation from holography
• Renormalization, Conformal Ward Identities and the Origin of a Conformal Anomaly Pole
• Dynamical response near quantum critical points
• Convergent Momentum-Space OPE and Bootstrap Equations in Conformal Field Theory
• A Scattering Amplitude in Conformal Field Theory
• Electromagnetic response of interacting Weyl semimetals
• Four-Point Functions in Momentum Space: Conformal Ward Identities in the Scalar/Tensor case
• Momentum space CFT correlators for Hamiltonian truncation
• Quantum critical response: from conformal perturbation theory to holography
• Comments on scale and conformal invariance in four dimensions
• TripleK: A Mathematica package for evaluating triple-K integrals and conformal correlation functions
• Holographic non-Gaussianities in general single-field inflation
• Anomalous Gravitational TTT Vertex, Temperature Inhomogeneity, and Pressure Anisotropy
• The Unique Polyakov Blocks
• The four-point correlation function of the energy-momentum tensor in the free conformal field theory of a scalar field

Cited by in corpus (29)

• On duality of color and kinematics in (A)dS momentum space
• Cosmological Scattering Equations
• Double copy structure of parity-violating CFT correlators
• Higher spin 3-point functions in 3d CFT using spinor-helicity variables
• Momentum space parity-odd CFT 3-point functions
• Cosmological Scattering Equations at Tree-level and One-loop
• Relation between parity-even and parity-odd CFT correlation functions in three dimensions
• Constraining momentum space correlators using slightly broken higher spin symmetry
• A handbook of holographic 4-point functions
• Yangian Bootstrap for Massive Feynman Integrals
• Effective Field Theories and Cosmological Scattering Equations
• Constraining momentum space CFT correlators with consistent position space OPE limit and the collider bound
• Enhanced Soft Limits in de Sitter Space
• Momentum space CFT correlators of non-conserved spinning operators
• All plus four point (A)dS graviton function using generalized on-shell recursion relation
• Conformal partial waves in momentum space
• Notes on conformal anomaly, nonlocal effective action and the metamorphosis of the running scale
• Shift operators from the simplex representation in momentum-space CFT
• Four-point functions of gravitons and conserved currents of CFT in momentum space: testing the nonlocal action with the TTJJ
• BCJ Amplitude Relations for Anti-de Sitter Boundary Correlators in Embedding Space
• The Feyn-Structure of Yangian Symmetry
• Twist operators and pseudo entropies in two-dimensional momentum space
• Non-Local Symmetries of Planar Feynman Integrals
• $n$-point functions in Conformal Quantum Mechanics: A Momentum Space Odyssey
• On Gluing CFT Correlators to Higher-Spin Amplitudes in AdS
• Hypergeometry from $\hat{\mathrm{P}}$-Symmetry: Feynman Integrals in One and Two Dimensions
• The Conformal Anomaly Action to Fourth Order (4T) in $d=4$ in Momentum Space
• Nonperturbative dynamics of (2+1)d ${\varphi }^4$-theory from Hamiltonian truncation
• Anomalous Ward identities for on-shell amplitudes at the conformal fixed point