# Wilson line networks in $p$-adic AdS/CFT

## TL;DR

This paper proposes a lattice gauge theory formulation of the bulk in $p$-adic AdS/CFT using Wilson line networks on the Bruhat-Tits tree, successfully reproducing boundary correlation functions.

## Contribution

It introduces a novel lattice gauge theory approach to $p$-adic AdS/CFT, connecting Wilson line networks with boundary correlators.

## Key findings

- Wilson line networks reproduce all boundary correlation functions
- Bulk theory formulated as PGL$(2,Q_p)$ lattice gauge theory
- Establishes a concrete bulk-boundary correspondence in $p$-adic holography

## Abstract

The $p$-adic AdS/CFT is a holographic duality based on the $p$-adic number field $\mathbb{Q}_p$. For a $p$-adic CFT living on $\mathbb{Q}_p$ and with complex-valued fields, the bulk theory is defined on the Bruhat-Tits tree, which can be viewed as the bulk dual of $\mathbb{Q}_p$. We propose that bulk theory can be formulated as a lattice gauge theory of PGL$(2,\mathbb{Q}_p)$ on the Bruhat-Tits tree, and show that the Wilson line networks in this lattice gauge theory can reproduce all the correlation functions of the boundary $p$-adic CFT.