A multipoint conformal block chain in $d$ dimensions

Abstract

Conformal blocks play a central role in CFTs as the basic, theory-independent building blocks. However, only limited results are available concerning multipoint blocks associated with the global conformal group. In this paper, we systematically work out the $d$-dimensional $n$-point global conformal blocks (for arbitrary $d$ and $n$) for external and exchanged scalar operators in the so-called comb channel. We use kinematic aspects of holography and previously worked out higher-point AdS propagator identities to first obtain the geodesic diagram representation for the $(n+2)$-point block. Subsequently, upon taking a particular double-OPE limit, we obtain an explicit power series expansion for the $n$-point block expressed in terms of powers of conformal cross-ratios. Interestingly, the expansion coefficient is written entirely in terms of Pochhammer symbols and $(n-4)$ factors of the generalized hypergeometric function ${}_3{F}_2$, for which we provide a holographic explanation. This generalizes the results previously obtained in the literature for $n=4,5$. We verify the results explicitly in embedding space using conformal Casimir equations.

arXiv comment35 pages + appendices + references, several figures. Mathematica notebook containing the main result included as an ancillary file

References in corpus (58)

• Gauge Theory Correlators from Non-Critical String Theory
• The Large N Limit of Superconformal Field Theories and Supergravity
• Bounding scalar operator dimensions in 4D CFT
• 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
• The Analytic Bootstrap and AdS Superhorizon Locality
• Spinning Conformal Correlators
• Analyticity in Spin in Conformal Theories
• Convexity and Liberation at Large Spin
• Universality of Long-Distance AdS Physics from the CFT Bootstrap
• Projectors, Shadows, and Conformal Blocks
• Bootstrapping the O(N) Vector Models
• The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT
• Spinning Conformal Blocks
• A Natural Language for AdS/CFT Correlators
• Large Spin Perturbation Theory
• Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks
• Towards Feynman rules for Mellin amplitudes in AdS/CFT
• Weight Shifting Operators and Conformal Blocks
• $20^{\prime }$ Five-Point Function from $Ad{S}_5\times S^5$ Supergravity
• Recursion Relations for Conformal Blocks
• Universal anomalous dimensions at large spin and large twist
• Analytic bootstrap at large spin
• Multipoint Conformal Blocks in the Comb Channel
• Superintegrability of $d$-dimensional Conformal Blocks
• Spinning Witten Diagrams
• $p$-adic AdS/CFT
• Projectors and seed conformal blocks for traceless mixed-symmetry tensors
• Treelike interactions and fast scrambling with cold atoms
• Radial expansion for spinning conformal blocks
• Seed Conformal Blocks in 4D CFT
• On Feynman rules for Mellin amplitudes in AdS/CFT
• Harmony of Spinning Conformal Blocks
• Casimir recursion relations for general conformal blocks
• Edge length dynamics on graphs with applications to $p$-adic AdS/CFT
• Holographic dual of the five-point conformal block
• Witten Diagrams for Torus Conformal Blocks
• Anatomy of Geodesic Witten Diagrams
• Holographic conformal blocks from interacting Wilson lines
• Holographic Conformal Partial Waves as Gravitational Open Wilson Networks
• Geodesic Diagrams, Gravitational Interactions & OPE Structures
• AdS Weight Shifting Operators
• Recursion relation for general 3d blocks
• O(N) and O(N) and O(N)
• Spinning Geodesic Witten Diagrams
• Geodesic Witten diagrams with an external spinning field
• Fermions in Geodesic Witten Diagrams
• Scalar Blocks as Gravitational Wilson Networks
• Non-local non-linear sigma models
• Superconformal Blocks: General Theory
• Scalar fields on $p$AdS
• Wilson line networks in $p$-adic AdS/CFT
• Comments on spinning OPE blocks in AdS${}_3$/CFT${}_2$
• Holography on local fields via Radon Transform
• Enhanced Symmetry of the $p$-adic Wavelets
• Mixed field theory

Cited by in corpus (23)

• Multipoint Bootstrap I: Light-Cone Snowflake OPE and the WL Origin
• Scattering Equations in AdS: Scalar Correlators in Arbitrary Dimensions
• Gaudin Models and Multipoint Conformal Blocks: General Theory
• From Gaudin Integrable Models to $d$-dimensional Multipoint Conformal Blocks
• Higher-Point Conformal Blocks in the Comb Channel
• On the Virasoro six-point identity block and chaos
• Gaudin Models and Multipoint Conformal Blocks II: Comb channel vertices in 3D and 4D
• Recursion relations for 5-point conformal blocks
• Seven-Point Conformal Blocks in the Extended Snowflake Channel and Beyond
• Six-Point Conformal Blocks in the Snowflake Channel
• Gaudin Models and Multipoint Conformal Blocks III: Comb channel coordinates and OPE factorisation
• Towards Feynman rules for conformal blocks
• Shadow Celestial Amplitude
• Lining up a Positive Semi-Definite Six-Point Bootstrap
• Improving the five-point bootstrap
• Feynman Rules for Scalar Conformal Blocks
• Efficient Rules for All Conformal Blocks
• Dimensional reduction of higher-point conformal blocks
• Conformal partial waves in momentum space
• Bending the Bruhat-Tits Tree I:Tensor Network and Emergent Einstein Equations
• Stampedes II: Null Polygons in Conformal Gauge Theory
• Conformal Blocks in Two and Four Dimensions from Oscillator Representations
• On-shell Correlators and Color-Kinematics Duality in Curved Symmetric Spacetimes