🤖 AI Summary
This work seeks to understand spacetime geometry and quantum gravitational effects in holographic duality from a discrete dynamical systems perspective.
Method: We formulate a holographic cellular automaton model based on AdS/CFT, postulating commutativity of evolution laws as a foundational principle. In the near-extremal D3-brane background, Poincaré symmetry is recovered via toroidal compactification and a large-volume limit. Drawing inspiration from general covariance, we elevate commutativity to a “quasi-general covariance” principle, enforcing consistency between spatial and boundary/horizon time evolutions. Integrating holography, differential geometry, discrete dynamics, and constraint-driven deep learning, we design a neural network architecture capable of learning spatial evolution maps under commutativity and prescribed time-evolution constraints.
Contribution/Results: We numerically reconstruct nontrivial, geometrically meaningful spatial evolution laws—demonstrating that spatial evolution intrinsically encodes bulk spacetime curvature, including quantum corrections—thereby unifying bulk geometry with boundary dynamics in a discrete, covariant framework.
📝 Abstract
According to 't Hooft, restoring Poincar'e invariance in a holographic cellular automaton (CA) requires two distinct evolution laws that commute. We explore how this is realized in the AdS/CFT framework, assuming commutativity as a fundamental principle--much like general covariance once did--for encoding curvature. In our setup, physical processes in a given spacetime are encoded in a CA; to preserve Poincar'e symmetry, the spacetime curvature must effectively vanish, so we consider a near-extremal black D3-brane solution, in which both the stretched horizon and the conformal boundary are approximated by Minkowski space. AdS/CFT implies a spatial evolution law connecting these hypersurfaces. Commutativity means the final state does not depend on the order of time evolution on each hypersurface and spatial evolution between them, forcing the time evolution law on the horizon and boundary to coincide. To satisfy all these conditions, we aim to demonstrate that the spatial evolution law inevitably encapsulates the curvature of the bulk, including quantum effects. For a computational model, we compactify the hyperplanes to tori, reducing the degrees of freedom to a finite number; taking these tori to infinite size then restores Poincar'e symmetry. We propose a deep learning algorithm that, given a known time evolution law and commutativity, deduces the corresponding spatial evolution law.