The Arithmetic Geometry of AdS₂ and its Continuum Limit
According to the 't Hooft-Susskind holography, the black hole entropy, 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 = exp(BH). In previous work, we have proposed that the near-horizon...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2021 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2021
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/211184 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | The Arithmetic Geometry of AdS₂ and its Continuum Limit. Minos Axenides, Emmanuel Floratos and Stam Nicolis. SIGMA 17 (2021), 004, 22 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862622233355091968 |
|---|---|
| author | Axenides, Minos Floratos, Emmanuel Nicolis, Stam |
| author_facet | Axenides, Minos Floratos, Emmanuel Nicolis, Stam |
| citation_txt | The Arithmetic Geometry of AdS₂ and its Continuum Limit. Minos Axenides, Emmanuel Floratos and Stam Nicolis. SIGMA 17 (2021), 004, 22 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | According to the 't Hooft-Susskind holography, the black hole entropy, 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 = exp(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 AdS₂[ℤ] discrete, finite, and random geometry, where ∝ 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, SBH. What has been left as an open problem is how the smooth AdS₂ geometry can be recovered, in the limit when → ∞. In the present article, we solve this problem by showing that the discrete and finite AdS₂[ℤ] geometry can be embedded in a family of finite geometries, AdSᴹ₂[ℤ], 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, and can be understood as ''infrared'' and ''ultraviolet'' cutoffs, respectively. The above construction enables us to obtain the continuum limit of the AdSᴹ₂[ℤ] discrete and finite geometry, by taking both and 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[ℤ] 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.
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| first_indexed | 2026-03-14T13:33:17Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211184 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T13:33:17Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Axenides, Minos Floratos, Emmanuel Nicolis, Stam 2025-12-25T13:25:11Z 2021 The Arithmetic Geometry of AdS₂ and its Continuum Limit. Minos Axenides, Emmanuel Floratos and Stam Nicolis. SIGMA 17 (2021), 004, 22 pages 1815-0659 2020 Mathematics Subject Classification: 14L35; 11D45; 83C57 arXiv:1908.06641 https://nasplib.isofts.kiev.ua/handle/123456789/211184 https://doi.org/10.3842/SIGMA.2021.004 According to the 't Hooft-Susskind holography, the black hole entropy, 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 = exp(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 AdS₂[ℤ] discrete, finite, and random geometry, where ∝ 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, SBH. What has been left as an open problem is how the smooth AdS₂ geometry can be recovered, in the limit when → ∞. In the present article, we solve this problem by showing that the discrete and finite AdS₂[ℤ] geometry can be embedded in a family of finite geometries, AdSᴹ₂[ℤ], 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, and can be understood as ''infrared'' and ''ultraviolet'' cutoffs, respectively. The above construction enables us to obtain the continuum limit of the AdSᴹ₂[ℤ] discrete and finite geometry, by taking both and 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[ℤ] 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. This work spanned many places and benefited from discussions with many people. We would like to thank, in particular, Costas Bachas and John Iliopoulos at the LPTENS, Gia Dvali, Alex Kehagias, Boris Pioline, Kyriakos Papadodimas, and Eliezer Rabinovici at CERN. We acknowledge the warm hospitality at Ecole Normale Supérieure, Paris, the Theory Division at CERN, and the Institute of Nuclear and Particle Physics of the NRCPS Demokritos. We would also wish to thank Professor H.W. Lenstra for illuminating correspondence and the referees of our paper for the interest they showed in our submission and their detailed reports, which allowed us to sharpen our arguments and improve the presentation. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Arithmetic Geometry of AdS₂ and its Continuum Limit Article published earlier |
| spellingShingle | The Arithmetic Geometry of AdS₂ and its Continuum Limit Axenides, Minos Floratos, Emmanuel Nicolis, Stam |
| title | The Arithmetic Geometry of AdS₂ and its Continuum Limit |
| title_full | The Arithmetic Geometry of AdS₂ and its Continuum Limit |
| title_fullStr | The Arithmetic Geometry of AdS₂ and its Continuum Limit |
| title_full_unstemmed | The Arithmetic Geometry of AdS₂ and its Continuum Limit |
| title_short | The Arithmetic Geometry of AdS₂ and its Continuum Limit |
| title_sort | arithmetic geometry of ads₂ and its continuum limit |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211184 |
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