Krylov Subspace Dynamics as Near-Horizon AdS$_2$ Holography

TL;DR

This paper establishes a holographic duality between Krylov subspace operator dynamics and near-horizon AdS$_2$ gravity, revealing a geometric interpretation of operator growth and chaos saturation.

Contribution

It introduces a holographic framework linking Krylov subspace evolution to AdS$_2$ gravity, including a new Krylov-based holographic dictionary and stability conditions.

Findings

Deep interior of Krylov subspace maps to AdS$_2$ near-horizon geometry.

Linear growth rate of Lanczos coefficients equals Hawking temperature, saturating chaos bound.

Breitenlohner-Freedman bound emerges as a stability criterion in the dual description.

Abstract

We establish a holographic gravitational dual for the fundamental dynamical equations governing operator growth in Krylov subspace. Specifically, we show that the deep interior of the Krylov subspace maps directly to the near-horizon regime of AdS$_2$ gravity. We demonstrate that, in the continuum limit, the discrete evolution on the Krylov chain transforms into the dynamics of a continuous field, which is isomorphic to the Klein-Gordon equation for a scalar field in the AdS$_2$ throat. This correspondence identifies the linear growth rate of Lanczos coefficients with the Hawking temperature, $\alpha=\pi T$, thereby recovering the saturation of the maximal chaos bound. Notably, the Breitenlohner-Freedman bound, a fundamental stability criterion in AdS gravity, emerges as a necessary consistency requirement for the dual description of Krylov subspace dynamics. Our results advance a Krylov-based holographic dictionary in a unified $SL(2, \mathbb{R})$ representation, revealing that the emergent geometry of Krylov subspace is a reflection of the near-horizon AdS spacetime.