Modular discretization of the AdS2/CFT1 Holography

TL;DR

This paper introduces a finite discretization of AdS2 geometry for extremal black holes using modular arithmetic, modeling dynamics with chaotic maps, and establishing a new holographic correspondence in a discretized setting.

Contribution

It proposes a novel discretization of AdS2 geometry using modular integers and models its dynamics with chaotic maps, establishing a new finite holographic duality.

Findings

Discretization of AdS2 as AdS2[N]=SL(2,Z_N)/SO(1,1,Z_N)

Modeling dynamics with generalized Arnol'd cat maps

Establishment of a finite AdS2/CFT1 holographic correspondence

Abstract

We propose a finite discretization for the black hole geometry and dynamics. We realize our proposal, in the case of extremal black holes, for which the radial and temporal near horizon geometry is known to be AdS$_2=SL(2,\mathbb{R})/SO(1,1,\mathbb{R})$. We implement its discretization by replacing the set of real numbers $\mathbb{R}$ with the set of integers modulo $N$, with AdS$_2$ going over to the finite geometry AdS$_2[N]=SL(2,\mathbb{Z}_N)/SO(1,1,\mathbb{Z}_N)$. We model the dynamics of the microscopic degrees of freedom by generalized Arnol'd cat maps, ${\sf A}\in SL(2,\mathbb{Z}_N)$, which are isometries of the geometry at both the classical and quantum levels. These exhibit well studied properties of strong arithmetic chaos, dynamical entropy, nonlocality and factorization in the cutoff discretization $N$, which are crucial for fast quantum information processing. We construct, finally, a new kind of unitary and holographic correspondence, for AdS$_2[N]$/CFT$_1[N]$, via coherent states of both the bulk and boundary geometries.