Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 004, 22 pages      arXiv:1908.06641      https://doi.org/10.3842/SIGMA.2021.004

The Arithmetic Geometry of AdS$_2$ and its Continuum Limit

Minos Axenides a, Emmanuel Floratos ab and Stam Nicolis c
a) Institute of Nuclear and Particle Physics, NCSR ''Demokritos'', Aghia Paraskevi, GR-15310, Greece
b) Physics Department, University of Athens, Zografou University Campus, Athens, GR-15771, Greece
c) Institut Denis Poisson, Université de Tours, Université d'Orléans, CNRS (UMR7013), Parc Grandmont, 37200 Tours, France

Received April 02, 2020, in final form January 02, 2021; Published online January 09, 2021

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 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, $S_\mathrm{BH}$. 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, we show how it is possible to recover the continuous, toroidally compactified, AdS$_2[\mathbb{Z}_N]$ 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 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.

Key words: arithmetic geometry of AdS$_2$; continuum limit of finite geometries; Fibonacci sequences.

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