Toward Krylov-based holography in double-scaled SYK

TL;DR

This work establishes a precise holographic dictionary between Krylov-space observables in the double-scaled SYK model and two-dimensional dilaton gravities. It shows that the growth rate of Krylov state complexity, $dC_S/dt$, is dual to the wormhole velocity, with coherent-state boundary diagnostics revealing firewall-like bulk reconstructions, and it equates an alternative bulk description to the proper momentum in early-time/low-energy regimes. It further extends the dictionary to higher-order and logarithmic Krylov complexities via replica-wormhole saddles and defines Krylov entropy as the von Neumann entropy of the parent geometry after tracing out baby universes in a third-quantized bulk. Collectively, these results position Krylov-space observables as sharp, versatile probes of bulk dynamics and topology in ensemble-averaged 2D gravity, with broad implications for holography and quantum chaos.

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.

Figures (3)

• Figure 1: (a) shows the backward shift in time of modes for bulk field $\phi$ according to equation (\ref{['eq-bulk mode shift']}). The blue curves are constant $r$ slices, and the red line is a constant time slice. (b) shows an effective picture with bulk modes unchanged, but shifted geometry. The green curve represents a perturbation. The orange dashed line stands for an infalling probe in the original geometry. The blue curve is the infalling probe in the effective geometry.
• Figure 2: (a) The schematic picture of replicated manifolds with three replicas in DSSYK. The crosses label the indices for the matrix $\hat{n}_{ij}(t)=(U^\dagger\hat{n}U)_{ij}$, which is no longer a diagonal matrix in general. The red lines indicate the contraction of indices. (b) The schematic picture of the disconnected saddle in the bulk gravity. (c) The schematic picture of the connected replica wormhole saddle in the bulk gravity.
• Figure 3: An illustration of the wavefunction $\psi_{L,b}(t)$ in Eq. \ref{['EQ377']}: the parent universe emits a baby universe of size b while its geodeaix boundary length $L$ evolves. The Euclidean half-disk at the bottom sets the Hartle-Hawking state, time evolution propagates it by $t/2$, the geodesic loop of length b marks the baby universe, and the geodesic boundary of length $L$ is the remaining parent.