Toward Krylov-based holography in double-scaled SYK

TL;DR

This paper develops a holographic dictionary linking Krylov complexity in the double-scaled SYK model to bulk gravitational phenomena, revealing new dualities and probes of bulk dynamics and topology.

Contribution

It establishes a precise duality between Krylov complexity growth and bulk wormhole velocity, connecting boundary complexity to bulk geometry and topology in 2D gravity.

Findings

Krylov complexity growth rate matches wormhole velocity.

Expectation value of Krylov complexity diagnoses firewall-like structures.

Higher-order Krylov complexities relate to replica wormholes.

Abstract

Building on the duality between Krylov complexity and geodesic length in Jackiw-Teitelboim and sine-dilaton gravity, we develop a precise holographic dictionary for quantities in the Krylov subspace of the double-scaled Sachdev-Ye-Kitaev model (DSSYK). First, we demonstrate that the growth rate of Krylov state complexity corresponds to the wormhole velocity, and show that its expectation value in coherent states serves as a boundary diagnostic of firewall-like structures via bulk reconstruction. We also delineate an alternative bulk description in terms of the proper momentum of an infalling particle at early times, establishing a threefold duality between the Krylov complexity growth rate, wormhole velocity, and proper momentum, with clear regimes of validity. Beyond the first moments, we argue that higher-order Krylov complexities capture connected bulk contributions encoded by replica wormholes, while the logarithmic variant probes the replica saddle structure. Finally, within a third-quantized setting incorporating baby universes, we show that the Krylov entropy equals the von Neumann entropy of the parent-geometry density matrix obtained after tracing out baby universes, thereby quantifying information flow into the baby universe sector. Together, these results elevate Krylov-space observables to sharp probes of bulk dynamics and topology in ensemble-averaged 2D gravity.