$p$-adic AdS/CFT

TL;DR

This work constructs and analyzes a $p$-adic version of AdS/CFT by replacing Euclidean space with the boundary $\mathbb{Q}_p$ extended to an unramified field $\mathbb{Q}_q$ and replacing the bulk with a Bruhat--Tits tree. Focusing on a single massive scalar field, it derives bulk-to-bulk and bulk-to-boundary propagators, relates the bulk mass to the scaling dimension via local $p$-adic zeta functions, and computes two-, three-, and four-point boundary correlators in closed form. The results reveal strong structural parallels with ordinary AdS/CFT when expressed through local zeta functions and highlight simplifications in $p$-adic amplitudes (notably for four-point functions) alongside a rich geometry of chordal distance and Wilson loops, suggesting a path toward an adelic holographic program. The discussion contrasts Archimedean and $p$-adic results and outlines future directions, including extensions to ramified cases, loop corrections, and dynamical bulk geometry.

Abstract

We construct a $p$-adic analog to AdS/CFT, where an unramified extension of the $p$-adic numbers replaces Euclidean space as the boundary and a version of the Bruhat-Tits tree replaces the bulk. Correlation functions are computed in the simple case of a single massive scalar in the bulk, with results that are strikingly similar to ordinary holographic correlation functions when expressed in terms of local zeta functions. We give some brief discussion of the geometry of $p$-adic chordal distance and of Wilson loops. Our presentation includes an introduction to $p$-adic numbers.

Figures (6)

• Figure 1: The Bruhat--Tits tree for $\mathbb{Q}_p$ with $p=2$, with a coordinate system $(z_0, z)$ shown which provides a useful parametrization of the bulk-to-boundary propagator.
• Figure 2: A variant of the Bruhat--Tits tree for the unramified extension $\mathbb{Q}_{p^2}$ with $p=2$.
• Figure 3: Left: The distance $d(a,b)$ between $a$ and $b$ is the number of steps along $T_q$ in the path from $a$ to $b$. The path from $x$ to $y$ in $T_q$ goes through $a$ and then $b$; likewise the path from $u$ to $v$. The intersection of the paths from $x$ to $y$ and from $u$ to $v$ is precisely the path from $a$ to $b$. Right: Paths on $T_q$ from three boundary points $x$, $y$, and $u$ meet at a unique bulk point $a$.
• Figure 4: (Color online.) Subway diagrams, indicating disjoint unions of paths on $T_q \sqcup \partial T_q$. (a) Paths from $x_1$, $x_2$, and $x_3$ meet at the bulk point $c \in T_q$ and comprise what we refer to as the main tree. The product $\prod_{i=1}^3 \hat{K}(a,x_i)$ relates to paths from the $x_i$ which all go to the point $a$ after first passing through the point $b$, which is the projection of $a$ onto the main tree. (b) and (c): Paths from $x_1$ and $x_2$ to $x_3$ and $x_4$ overlap between $c_1$ and $c_2$. There are two classes of subway diagrams contributing to the four-point amplitude, depending on whether the projection $b$ of $a$ onto the main trunk falls between $c_1$ and $c_2$ or on a leg between some $x_i$ and the appropriate $c_j$.
• Figure 5: (Color online.) Successive refinements of the Bruhat--Tits tree. The example shown is for $q=p=2$. (a) Successive refinements reveal more and more structure as we zoom in on any given bulk region. The first step is to write $T_q$ as a disjoint union of blocks $B$ as defined in (\ref{['Blocks']}). The next step is to write each block $B$ as a disjoint union of blocks $B_1$ as defined in (\ref{['BClasses']}); then each block $B_1$ is written as a disjoint union of blocks $B_2$, and so forth. (b) Successive refinements of $T_q$ lead to the enhanced tree $T_{qp}$, in which each vertex is a block $B_m$. The base tree $T_q$ is shown in gray, and the height $h$ measures how many steps away a point on $T_{qp}$ is from the base tree.