Modular discretization of the AdS2/CFT1 Holography

Abstract

We propose a finite discretization for the black hole 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 AdS${}_2=SL(2,\mathbb{R})/SO(1,1,\mathbb{R})$. We implement its discretization by replacing the set of real numbers $\mathbb{R}$ with the set of integers modulo $N$, with AdS${}_2$ going over to the finite geometry AdS${}_2[N]=SL(2,{\mathbb{Z}}_N)/SO(1,1,{\mathbb{Z}}_N)$. We model the dynamics of the microscopic degrees of freedom by generalized Arnol'd cat maps, $\mathsf{A}\in SL(2,{\mathbb{Z}}_N)$, which are isometries of the geometry at both the classical and quantum levels. These exhibit well studied properties of strong arithmetic chaos, dynamical entropy, nonlocality and factorization in the cutoff discretization $N$, which are crucial for fast quantum information processing. We construct, finally, a new kind of unitary and holographic correspondence, for AdS${}_2[N]$/CFT${}_1[N]$, via coherent states of both the bulk and boundary geometries.

33 pages LaTeX2e, 1 JPEG figure. Typos corrected, references added. Clarification of several points in the abstract and the text

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Cited by in corpus (24)

• State-Dependent Bulk-Boundary Maps and Black Hole Complementarity
• Comments on the Necessity and Implications of State-Dependence in the Black Hole Interior
• Quantum Simulation of Hyperbolic Space with Circuit Quantum Electrodynamics: From Graphs to Geometry
• Towards Explicit Discrete Holography: Aperiodic Spin Chains from Hyperbolic Tilings
• Protected $SL(2,\mathbb{R})$ Symmetry in Quantum Cosmology
• Breitenlohner-Freedman bound on hyperbolic tilings
• Smooth Causal Patches for AdS Black Holes
• Aperiodic spin chains at the boundary of hyperbolic tilings
• Klein four-group and Darboux duality in conformal mechanics
• The quantum cat map on the modular discretization of extremal black hole horizons
• The arithmetic geometry of AdS${}_2$ and its continuum limit
• A conformal group approach to the Dirac-Kähler system on the lattice
• Duality constructions from quantum state manifolds
• The continuum limit of the modular discretization of AdS${}_2$
• Chaotic Information Processing by Extremal Black Holes
• Wormhole-induced effective coupling in SYK chains
• Discrete JT gravity as an Ising model
• Cosmology as a CFT${}_1$
• Arnol'd cat map lattices
• Fast Scramblers, Democratic Walks and Information Fields
• Recent developments in the holographic description of quantum chaos
• Classical spin liquids from frustrated Ising models in hyperbolic space
• Construction of Metaplectic Representations of $S{L}_2({\mathbb{Z}}_{2^n})$ and Twisted Magnetic Translations
• Exponential mixing of all orders for Arnol'd cat map lattices