The continuum limit of the modular discretization of AdS${}_2$

Abstract

According to the holographic picture of 't Hooft and Susskind, the black hole entropy, $S_{\mathrm{BH}}$, is carried by the chaotic microscopic degrees of freedom, that live in the near horizon geometry 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 discrete, finite, random geometry, AdS${}_2[{\mathbb{Z}}_N]$, where $N$ is proportional to $S_{\mathrm{BH}}$. What had remained as an open problem was how the smooth AdS2 geometry can be recovered, in the limit when N goes to infinity. In this contribution we present the salient points of the solution to this problem, which involves embedding AdS${}_2[{\mathbb{Z}}_N]$ in a family of finite geometries, AdS${}_2^M[{\mathbb{Z}}_N]$, where $M$ is another integer, within 2+1 Minkowski spacetime. In this construction $N$ and $M$ can be considered IR and UV cutoffs. The continuum limit, corresponding to the smooth AdS${}_2$ geometry, is obtained by taking $N$ and $M$ to infinity in a correlated way, using properties of the Fibonacci and $k$-Fibonacci sequences. This method can be directly applied to higher-dimensional AdS spacetimes, also.

arXiv comment14 pages LaTeX, 3 figures. Uses PoS style files and JHEP BibTeX. Contribution to the Proceedings of the 2021 Corfu Workshops, "Elementary Particle Physics and Gravity", summarizing arXiv:1908.06641

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