PoS - Proceedings of Science
Volume 406 - Corfu Summer Institute 2021 "School and Workshops on Elementary Particle Physics and Gravity" (CORFU2021) - Workshop on Quantum Geometry, Field Theory and Gravity
The continuum limit of the modular discretization of AdS$_2$
E.G. Floratos*, M. Axenides and S. Nicolis
Full text: pdf
Published on: November 23, 2022
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
$N\propto S_\mathrm{BH}.$ What had remained as an open problem is how the smooth
AdS$_2$ geometry can be recovered, in the limit when $N\to\infty.$ 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.
DOI: https://doi.org/10.22323/1.406.0243
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