The Arithmetic Geometry of AdS₂ and its Continuum Limit

Abstract

According to the 't Hooft-Susskind holography, the black hole entropy, 𝑆BH, 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(𝑆BH). 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 AdS₂[ℤ𝑁] discrete, finite, and random geometry, where 𝑁 ∝ 𝑆BH. It has been constructed by purely arithmetic and group theoretical methods and was studied as a toy model for describing the dynamics of single particle probes of the near-horizon region of 4d extremal black holes, as well as to explain, in a direct way, the finiteness of the entropy, SBH. What has been left as an open problem is how the smooth AdS₂ geometry can be recovered, in the limit when 𝑁 → ∞. In the present article, we solve this problem by showing that the discrete and finite AdS₂[ℤ𝑁] geometry can be embedded in a family of finite geometries, AdSᴹ₂[ℤ𝑁], 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 AdSᴹ₂[ℤ𝑁] discrete and finite geometry, by taking both 𝑁 and 𝑀 to infinity in a specific correlated way, following a reverse process: Firstly, we show how it is possible to recover the continuous, toroidally compactified, AdS₂[ℤ𝑁] geometry by removing the ultraviolet cutoff; secondly, we show how it is possible to remove the infrared cutoff in a specific decompactification limit, while keeping the radius of AdS2 finite. It is in this way that we recover the standard non-compact AdS₂ continuum space-time. This method can be applied directly to higher-dimensional AdS spacetimes.