Learning geometries beyond asymptotic AdS

TL;DR

This paper introduces a machine learning approach for reconstructing bulk geometries in holography beyond asymptotic AdS, successfully applying it to models like SYK and identifying critical geometries.

Contribution

The authors develop a neural ODE-based method for bulk reconstruction from boundary data, extending holographic techniques to non-AdS spacetimes and applying it to the SYK model.

Findings

Successfully reconstructs AdS, Lifshitz, and hyperscaling geometries.

Identifies a critical curve for the SYK model where the geometry resembles an AdS$_2$ black hole.

Establishes an effective bulk dual of the SYK coupling.

Abstract

We present a data-driven method for holographic bulk reconstruction that works even when the spacetime is not asymptotically AdS. Given the data of boundary Green functions within a finite frequency window, we iteratively adjust a bulk metric with a finite radial cutoff until its holographic Green functions reproduce the boundary data. Based on the holographic Wilsonian renormalization group for the Klein-Gordon equation in an undetermined curve space, we construct a radial flow equation and transform it into a Neural ODE, which is an infinite-depth neural network for modeling continuous dynamics. Assuming the double-trace coupling $h$ in the Wilsonian action is real, we demonstrate that the Neural ODE can effectively learn the metrics with AdS, Lifshitz, and hyperscaling violated asymptotics. In particular, we apply the algorithm to the Sachdev-Ye-Kitaev (SYK) model which slightly deviates from the conformal limit. In the hyperparameter space spanned by the rescaled temperature $\bar{T}$ and the radial cutoff $\epsilon$, we identify a critical curve along which the learned metric is close to AdS$_2$ black hole with finite cutoff. We derive an approximate analytical expression for this curve, from which an effective bulk dual of the SYK coupling $v$ is established. Our work provides a promising way for using machine learning to depict the novel bulk geometry dual to the non-conformal boundary systems in the real world.