# Holographic dual of the five-point conformal block

## TL;DR

This paper introduces a holographic object that computes five-point conformal blocks in arbitrary dimensions, generalizing four-point diagrams, and proves its validity through eigenfunction analysis inspired by p-adic models.

## Contribution

It presents a novel holographic construction for five-point conformal blocks, extending previous four-point diagram methods and providing a new computational tool.

## Key findings

- The holographic object computes five-point conformal blocks in any dimension.
- The sum over geodesic bulk diagrams is an eigenfunction of the conformal Casimir.
- The approach is inspired by p-adic models on the Bruhat-Tits tree.

## Abstract

We present the holographic object which computes the five-point global conformal block in arbitrary dimensions for external and exchanged scalar operators. This object is interpreted as a weighted sum over infinitely many five-point geodesic bulk diagrams. These five-point geodesic bulk diagrams provide a generalization of their previously studied four-point counterparts. We prove our claim by showing that the aforementioned sum over geodesic bulk diagrams is the appropriate eigenfunction of the conformal Casimir operator with the right boundary conditions. This result rests on crucial inspiration from a much simpler $p$-adic version of the problem set up on the Bruhat-Tits tree.