$p$-adic AdS/CFT

TL;DR

This paper develops a $p$-adic version of AdS/CFT correspondence, replacing Euclidean space with $p$-adic numbers and the bulk with a Bruhat-Tits tree, and computes correlation functions analogous to the classical case.

Contribution

It introduces a novel $p$-adic holographic framework, connecting $p$-adic number theory with holography and computing correlation functions in this setting.

Findings

Correlation functions resemble classical holographic results when expressed with local zeta functions.

The geometry of $p$-adic chordal distance is discussed.

Brief discussion on Wilson loops in the $p$-adic context.

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.