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- W4287019162 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 $Npropto S_mathrm{BH}.$ What had remained as an open problem is how the smooth AdS$_2$ geometry can be recovered, in the limit when $Ntoinfty.$ In this contribution, we present the salient points of the solution to this problem, which involves embedding the discrete and finite AdS$_2[mathbb{Z}_N ]$ geometry 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. This construction allows 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." @default.
- W4287019162 created "2022-07-25" @default.
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- W4287019162 date "2022-11-23" @default.
- W4287019162 modified "2023-09-30" @default.
- W4287019162 title "The continuum limit of the modular discretization of AdS$_2$" @default.
- W4287019162 doi "https://doi.org/10.22323/1.406.0243" @default.
- W4287019162 hasPublicationYear "2022" @default.
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