# The arithmetic geometry of AdS$_2$ and its continuum limit

Abstract : According to the 't Hooft-Susskind holography, the black hole entropy is carried by microscopic degrees of freedom, which live in the near horizon region and have a Hilbert space of states of finite dimension $d=exp(S_\mathrm{BH})$. The AdS$_2[N]$ discrete and finite geometry, which has been constructed by purely arithmetic and group theoretical methods, was proposed as a toy model of the near horizon region of 4d extremal black holes, in order to describe the finiteness of the entropy, SBH, of these black holes. In the present article we show that, starting from the continuum 2d, anti-de Sitter geometry AdS$_2$, by an appropriate two-step process--discretization and toroidal compactification of the embedding 2+1 dimensional Minkowski space-time--we can derive a new construction of the finite AdS$_2[N]$ geometry. The above construction enables us to study the continuum limit of AdS$_2[N]$ as N goes to infinity, following a specific two-step, inverse, process: Firstly, we recover the continuous, toroidally compactified AdS$_2$ geometry; secondly, by taking an appropriate decompactification limit, we recover the standard non-compact AdS$_2$ continuum space-time.
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Journal articles
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https://hal.archives-ouvertes.fr/hal-02267951
Contributor : Stam Nicolis Connect in order to contact the contributor
Submitted on : Tuesday, August 20, 2019 - 9:43:19 AM
Last modification on : Tuesday, January 11, 2022 - 5:56:35 PM

### Identifiers

• HAL Id : hal-02267951, version 1
• ARXIV : 1908.06641

### Citation

Minos Axenides, Emmanuel Floratos, Stam Nicolis. The arithmetic geometry of AdS$_2$ and its continuum limit. Symmetry, Integrability and Geometry : Methods and Applications, National Academy of Science of Ukraine, 2021. ⟨hal-02267951⟩

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