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

Hyun-Sik Jeong

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, $α=πT$, 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.

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

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 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 throat. This correspondence identifies the linear growth rate of Lanczos coefficients with the Hawking temperature, , 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 representation, revealing that the emergent geometry of Krylov subspace is a reflection of the near-horizon AdS spacetime.
Paper Structure (11 equations, 1 figure, 1 table)

This paper contains 11 equations, 1 figure, 1 table.

Figures (1)

  • Figure 1: Holographic mapping between the AdS$_2$ bulk and the Krylov chain. The AdS$_2$ bulk spacetime, where a field $\Phi(t,\zeta)$ propagates towards the horizon, is mapped onto a semi-infinite chain with sites $\phi_n(t)$ in the large-$n$ limit: the continuum field $\phi(t,x)$ captures the near-horizon AdS$_2$ dynamics.