The arithmetic geometry of AdS${}_2$ and its continuum limit

Abstract

According to the 't Hooft-Susskind holography, the black hole entropy,$S_{\mathrm{BH}}$, is carried by the chaotic microscopic degrees of freedom, which live in the near horizon region and have a Hilbert space of states of finite dimension $d=\exp (S_{\mathrm{BH}})$. 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 AdS${}_2[{\mathbb{Z}}_N]$ discrete, finite and random geometry, where $N\propto S_{\mathrm{BH}}$. It has been constructed by purely arithmetic and group theoretical methods in order to explain, in a direct way, the finiteness of the entropy, $S_{\mathrm{BH}}$. What has been left as an open problem is how the smooth AdS${}_2$ geometry can be recovered, in the limit when $N\to \infty$. In the present article we solve this problem, by showing that the discrete and finite AdS${}_2[{\mathbb{Z}}_N]$ geometry can be embedded in a family of finite geometries, AdS${}_2^M[{\mathbb{Z}}_N]$, where $M$ is another integer. This family can be constructed by an appropriate toroidal compactification and discretization of the ambient $(2+1)$-dimensional Minkowski space-time. In this construction $N$ and $M$ can be understood as "infrared" and "ultraviolet" cutoffs respectively. The above construction enables us to obtain the continuum limit of the AdS${}_2^M[{\mathbb{Z}}_N]$ discrete and finite geometry, by taking both $N$ and $M$ to infinity in a specific correlated way, following a reverse process: Firstly, by recovering the continuous, toroidally compactified, AdS${}_2[{\mathbb{Z}}_N]$ geometry by removing the ultraviolet cutoff; secondly, by removing the infrared cutoff in a specific decompactification limit, while keeping the radius of AdS${}_2$ finite. It is in this way that we recover the standard non-compact AdS${}_2$ continuum space-time. This method can be applied directly to higher-dimensional AdS spacetimes.

22 pages, LaTeX2e, many PNG figures. v1: Uses utphys.bst for the references. v2: Clarifications about the precursors, additional figures and references. v3: Further clarifying remarks and references. v4: Streamlined presentation; references added. v5: Further improvements of the presentation, references added. v6: Final version, as published in SIGMA. The displayed abstract is shortened

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• Arnol'd cat map lattices
• Toward Holography on Biregular Trees
• Exponential mixing of all orders for Arnol'd cat map lattices