On Conformal Block, Crossing Kernel and Multi-variable Hypergeometric Functions

Abstract

In this note, we present an alternative representation of the conformal block with external scalars in general spacetime dimensions in terms of a finite summation over Appell fourth hypergeometric function ${\mathbf{F}}_4$. We also construct its generalization to the non-local primary exchange operator with continuous spin and its corresponding Mellin representation which are relevant for Lorentzian spacetime. Using these results we apply the Lorentzian inversion formula to compute so-called crossing kernel in general spacetime dimensions, the resultant expression can be written as a double infinite summation over certain Kampe de Feriet hypergeometric functions with the correct double trace operator singularity structures. We also include some complementary computations in AdS space, demonstrating the orthogonality of conformal blocks and performing the decompositions.

37 pages, 7 figures. Various references added and typos corrected

References in corpus (7)

• Spinning AdS Propagators
• $d$-dimensional SYK, AdS Loops, and $6j$ Symbols
• On the Polyakov-Mellin bootstrap
• Recursion Relations in Witten Diagrams and Conformal Partial Waves
• On Some Hypergeometric Solutions of the Conformal Ward Identities of Scalar 4-point Functions in Momentum Space
• The General 3-Graviton Vertex ($TTT$) of Conformal Field Theories in Momentum Space in $d=4$
• Scalar fields on λ-deformed cosets

Cited by in corpus (9)

• Selected Topics in Analytic Conformal Bootstrap: A Guided Journey
• Four-Point Functions in Momentum Space: Conformal Ward Identities in the Scalar/Tensor case
• Lightcone expansions of conformal blocks in closed form
• The Inversion Formula and 6j Symbol for 3d Fermions
• Regge OPE blocks and light-ray operators
• Superconformal Block from Holographic Geometry
• Recursive structure of the Gauß hypergeometric function and boundary/crosscap conformal block
• Exact 3D Conformal Blocks from Fractional Calculus
• The Conformal Anomaly Action to Fourth Order (4T) in $d=4$ in Momentum Space