# Geodesic bulk diagrams on the Bruhat-Tits tree

## TL;DR

This paper extends the concept of geodesic bulk diagrams to the p-adic AdS/CFT setting using the Bruhat-Tits tree, revealing simplifications and universal structures in four-point functions and proposing a minimal bulk action for free boundary theories.

## Contribution

It demonstrates the duality of geodesic bulk diagrams in p-adic AdS/CFT and introduces a minimal bulk action on the Bruhat-Tits tree that reproduces key boundary correlators.

## Key findings

- Derivatives vanish in the conformal block decomposition.
- Universal forms of coefficients and structure constants in p-adic and real cases.
- A minimal bulk action reproduces free boundary correlators.

## Abstract

Geodesic bulk diagrams were recently shown to be the geometric objects which compute global conformal blocks. We show that this duality continues to hold in $p$-adic AdS/CFT, where the bulk is replaced by the Bruhat-Tits tree, an infinite regular graph with no cycles, and the boundary is described by $p$-adic numbers, rather than reals. We apply the duality to evaluate the four-point function of scalar operators of generic dimensions using tree-level bulk diagrams. Relative to standard results from the literature, we find intriguing similarities as well as significant simplifications. Notably, all derivatives disappear in the conformal block decomposition of the four-point function. On the other hand, numerical coefficients in the four-point function as well as the structure constants take surprisingly universal forms, applicable to both the reals and the $p$-adics when expressed in terms of local zeta functions. Finally, we present a minimal bulk action with nearest neighbor interactions on the Bruhat-Tits tree, which reproduces the two-, three-, and four-point functions of a free boundary theory.