Holographic Krylov Complexity for Conformal Quiver Gauge Theories

We investigate holographic Krylov complexity in fully top-down AdS$_3$ and AdS$_2$ supergravity backgrounds dual to two-dimensional linear-quiver SCFTs and one-dimensional conformal quantum mechanics. In these geometries, the warp factors, dilaton and other fields depend non-trivially on the ‘quiver coordinate’ (denoted by $η$ in this paper). This $η$-coordinate encodes the color and flavor data of the dual theories. As a consequence, a massive probe following a holographic geodesic necessarily moves simultaneously in the radial AdS direction and along the ‘quiver direction’. This produces new contributions to the proper momentum and hence to the rate of Krylov complexity growth, which is absent in bottom-up AdS models. We show that the $η$-motion is generically damped, with a time-scale governed by the UV cutoff of the geodesic problem, and modifies the early-time evolution of complexity in a quiver-dependent way. At late times, the $η$-dynamics freezes and the growth becomes universal, matching pure Poincare AdS predictions. Studying Abelian and non-Abelian T-dual backgrounds of AdS$_3\times S^3\times T^4$, quivers with localized flavor groups, and quivers with smeared flavor groups, we quantify how quiver parameters shape the operator-spreading dynamics. Our results provide a systematic characterization of Krylov complexity in top-down AdS$_3$/AdS$_2$ duals and reveal a holographic mechanism through which complexity probes both ultraviolet quiver structure and emergent infrared universality.

💡 Research Summary

This paper presents a systematic study of holographic Krylov complexity in fully top‑down AdS₃ and AdS₂ supergravity backgrounds that are dual to two‑dimensional linear‑quiver superconformal field theories (SCFTs) and one‑dimensional conformal quantum mechanics, respectively. The key novelty of these backgrounds is the presence of a “quiver coordinate” η on which the warp factors, dilaton, and internal fluxes depend non‑trivially. The functions h₄(η) and h₈(η) encode the ranks of gauge and flavor groups in the quiver; they are piecewise‑linear (polygonal) and contain localized sources at the points where the quiver changes.

A massive probe particle of mass m is introduced, and its world‑line action is written in the Einstein frame. Because the metric depends on η, the particle cannot fall purely radially; its trajectory involves both the AdS radial coordinate r(t) and the quiver coordinate η(t). The induced world‑line metric leads to coupled equations of motion. The η‑equation contains an effective potential derived from the gradients of h₄ and h₈ and a damping term proportional to the UV regulator ε that controls the initial conditions. Consequently, η‑motion is significant only for early times t ≲ t_* ∼ ε⁻¹ and then freezes, leaving a purely radial infall at late times.

Krylov complexity C(t) is defined via the relation \dot C(t)=−ε P_{\bar ρ}, where P_{\bar ρ} is the proper momentum of the probe. In these top‑down models the proper momentum has two contributions: the usual radial component P_r = m e^{-Φ}A(η) \dot r and an additional quiver component P_η = m e^{-Φ}A(η)³ \dot η. The η‑contribution yields extra terms in \dot C that depend on the slopes of h₄ and h₈, i.e., on the quiver parameters β_i and ν_i. Thus the early‑time growth of Krylov complexity directly probes the ultraviolet quiver data.

The authors solve the coupled equations numerically for four representative families of backgrounds: (i) the Abelian T‑dual of AdS₃×S³×T⁴, (ii) the non‑Abelian T‑dual of the same geometry, (iii) quivers with localized flavor “kinks”, and (iv) quivers with smeared flavor distributions. In each case the initial η‑motion produces a quiver‑dependent deviation from the universal AdS growth rate λ/(2π). Near the kink points the deviation is most pronounced, reflecting the abrupt change in gauge‑flavor ranks. At later times the η‑velocity decays, the quiver contribution vanishes, and the growth rate settles to the universal value dictated solely by the AdS radius, in agreement with bottom‑up proposals.

The paper concludes that (1) top‑down warped AdS backgrounds naturally incorporate quiver dynamics absent in bottom‑up models, (2) Krylov complexity serves as a sensitive diagnostic of UV quiver structure through its early‑time behavior, and (3) the late‑time universality of complexity growth is robust, confirming expectations from holographic chaos and operator growth. The authors suggest extensions to more general quiver configurations, multi‑particle probes, and connections to other quantum information measures such as out‑of‑time‑order correlators and entanglement entropy.