The continuum limit of the modular discretization of AdS$_2$

TL;DR

This paper demonstrates how the smooth AdS$_2$ geometry emerges as a continuum limit of a discrete, finite, random geometry model, using a correlated limit involving Fibonacci sequences, with potential applications to higher-dimensional AdS spaces.

Contribution

It provides a method to recover smooth AdS$_2$ geometry from a discrete model by taking a correlated limit of parameters, extending previous finite geometry models.

Findings

Continuum limit recovers smooth AdS$_2$ geometry.

Uses Fibonacci sequences to correlate IR and UV cutoffs.

Method applicable to higher-dimensional AdS spacetimes.

Abstract

According to the holographic picture of 't Hooft and Susskind, the black hole entropy, $S_{\rm BH}$, is carried by the chaotic microscopic degrees of freedom, that live in the near horizon geometry and have a Hilbert space of states of finite dimension, $d=\exp(S_{\rm BH})$. In previous work we have proposed that the near horizon geometry, when the microscopic degrees of freedom can be resolved, can be described by the discrete, finite, random geometry, AdS$_2[\mathbb{Z}_N]$, where $N$ is proportional to $S_{\rm BH}$. What had remained as an open problem was how the smooth AdS2 geometry can be recovered, in the limit when N goes to infinity. In this contribution we present the salient points of the solution to this problem, which involves embedding AdS$_2[\mathbb{Z}_N]$ in a family of finite geometries, AdS$_2^M[\mathbb{Z}_N]$, where $M$ is another integer, within 2+1 Minkowski spacetime. In this construction $N$ and $M$ can be considered IR and UV cutoffs. The continuum limit, corresponding to the smooth AdS$_2$ geometry, is obtained by taking $N$ and $M$ to infinity in a correlated way, using properties of the Fibonacci and $k$-Fibonacci sequences. This method can be directly applied to higher-dimensional AdS spacetimes, also.