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Register a new userThe arithmetic geometry of AdS$_2$ and its continuum limit
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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 $Npropto S_mathrm{BH}$. 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 $Ntoinfty$. 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.
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