Modular discretization of the AdS2/CFT1 holography

We propose a finite discretization for the black hole, near horizon, geometry and dynamics. We realize our proposal, in the case of extremal black holes, for which the radial and temporal near horizon geometry is known to be \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathrm{Ad}{{\mathrm{S}}_2}={{{\mathrm{S}\mathrm{L}\left( {2,\mathbb{R}} \right)}} \left/ {{\mathrm{S}\mathrm{O}\left( {1,1,\mathbb{R}} \right)}} \right.} $\end{document}. We implement its discretization by replacing the set of real numbers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathbb{R} $\end{document} with the set of integers modulo N with AdS2 going over to the finite geometry \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathrm{Ad}{{\mathrm{S}}_2}\left[ N \right]={{{\mathrm{S}\mathrm{L}\left( {2,{{\mathbb{Z}}_N}} \right)}} \left/ {{\mathrm{S}\mathrm{O}\left( {1,1,{{\mathbb{Z}}_N}} \right)}} \right.} $\end{document}. We model the dynamics of the microscopic degrees of freedom by generalized Arnol’d cat maps, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathrm{A}\in \mathrm{SL}\left( {2,{{\mathbb{Z}}_N}} \right) $\end{document} which are isometries of the geometry, at both the classical and quantum levels. These are well known to exhibit fast quantum information processing through the well studied properties of strong arithmetic chaos, dynamical entropy, nonlocality and factorization in the cutoff discretization N. We construct, finally, a new kind of unitary and holographic correspondence, for AdS2[N]/CFT1[N], via coherent states of the bulk and boundary geometries.