The arithmetic geometry of AdS$_2$ and its continuum limit
Minos Axenides, Emmanuel Floratos, Stam Nicolis
备注：
22 pages, LaTeX2e, many PNG figures. v1: Uses utphys.bst for the references. v2: Clarifications about the precursors, additional figures and references. v3: Further clarifying remarks and references. v4: Streamlined presentation; references added. v5: Further improvements of the presentation, references added. v6: Final version, as published in SIGMA. The displayed abstract is shortened
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 $N\propto S_\mathrm{BH}$. It has been constructed by purely arithmetic and group theoretical methods in order to explain, in a direct way, the finiteness of the entropy, $S_\mathrm{BH}$. What has been left as an open problem is how the smooth AdS$_2$ geometry can be recovered, in the limit when $N\to\infty$. In the present article we solve this problem, by showing that the discrete and finite AdS$_2[\mathbb{Z}_N]$ geometry can be embedded 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. The above construction enables 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.
Confinement Versus Screening in the Schwinger Model on AdS$_2$ from Bosonization and Tensor Networks
University of California, Los Angeles（加州大学洛杉矶分校）；Mani L. Bhaumik Institute for Theoretical Physics, University of California, Los Angeles（加州大学洛杉矶分校马尼·L·巴乌米克理论物理研究所）；Center for Quantum Science and Engineering, University of California, Los Angeles（加州大学洛杉矶分校量子科学与工程中心）
