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

## TL;DR

This paper introduces a new finite-sum representation of conformal blocks using Appell hypergeometric functions, generalizes to non-local operators with continuous spin, and applies these results to compute crossing kernels in various spacetime dimensions.

## Contribution

It provides an alternative finite-sum representation of conformal blocks and extends the crossing kernel computation to general spacetime dimensions using hypergeometric functions.

## Key findings

- Finite-sum representation of conformal blocks with Appell functions
- Generalization to non-local primary exchange operators with continuous spin
- Explicit computation of crossing kernels in general spacetime dimensions

## 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 ${\bf{F}}_4$. 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 Kamp\~{e} de F\~{e}riet 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.