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arXiv:2610.13555

Holographic Krylov complexity in confining backgrounds: particle, string, and frames

Jasper Nongmaithem, Deepak Kumar Kar, Ayush Kumar, Shreeya Rajesh Parab, David Dudal, Subhash Mahapatra

hep-thgr-qchep-phquant-phSubmitted Oct 9, 2026• Announced Oct 12, 2026
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Abstract

We investigate holographic Krylov complexity in bottom-up holographic QCD models describing the confined phase, using the recently proposed correspondence between the growth rate of Krylov complexity and the proper radial momentum of probes, such as particles and strings, falling in the bulk spacetime. As representative confining backgrounds, we consider two horizonless geometries, namely the dilaton-deformed AdS-soliton and the deformed thermal-AdS spacetime, both arising as solutions of an Einstein-dilaton gravity system. We find that the Krylov complexity associated with a massive particle exhibits qualitatively distinct behavior in these two backgrounds when analyzed in the Einstein frame. In the deformed AdS-soliton geometry, the complexity undergoes periodic oscillations, with both the oscillation period and amplitude depending on the parameters of the model. In contrast, in the deformed thermal-AdS geometry, the complexity grows exponentially at late times and does not display oscillatory behavior. We then investigate string dynamics in the string frame and find that the Krylov complexity exhibits oscillatory behavior in both confining backgrounds. This behavior is traced to the emergence of an additional turning point in the deep infrared of the string-frame geometry. Motivated by this observation, we further analyze massive-particle dynamics in the string frame, where we find qualitatively similar behavior in the two confining backgrounds. Our results demonstrate that the Krylov complexity of confined holographic systems depends sensitively on the choice of probe and frame, while also revealing a closer correspondence between the two confining geometries in the string-frame description.

Comments
36 pages, 12 figures, comments are welcome

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The boundary quantity that captures the growth of black-hole interiors quantified by holographic complexity remains unknown beyond 2d dilaton gravity. We provide a critical analysis of a partition-function construction of Krylov spread complexity for thermofield-double states that provides a dimension-independent boundary reconstruction from semiclassical holographic partition functions, while developing the present dynamical and bulk construction for the BTZ saddle. In the double-scaled Sachdev-Ye-Kitaev model, where exact and semiclassical results can be compared, we show that the classical limit is reliable only when taken after the complexity has been reconstructed; taking this limit at the level of individual Lanczos coefficients discards essential information. Applying the construction to a large-central-charge two-dimensional conformal field theory above the Hawking-Page temperature dual to a Bañados-Teitelboim-Zanelli black hole, we find an intermediate departure from early-time quadratic growth followed by behavior compatible with a return toward asymptotically quadratic growth, rather than the linear late-time behavior of the volume and the standard finite-functional complexity = anything class. We then match this boundary behavior to a generalized complexity = anything bulk object built from an infinite series of extrinsic-curvature invariants. The construction provides a systematic route from black-hole thermodynamics to Krylov dynamics and can naturally be extended to higher-dimensional holographic black holes.

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We investigate holographic Krylov complexity in bottom-up holographic QCD models describing the confined phase, using the recently proposed correspondence between the growth rate of Krylov complexity and the proper radial momentum of probes, such as particles and strings, falling in the bulk spacetime. As representative confining backgrounds, we consider two horizonless geometries, namely the dilaton-deformed AdS-soliton and the deformed thermal-AdS spacetime, both arising as solutions of an Einstein-dilaton gravity system. We find that the Krylov complexity associated with a massive particle exhibits qualitatively distinct behavior in these two backgrounds when analyzed in the Einstein frame. In the deformed AdS-soliton geometry, the complexity undergoes periodic oscillations, with both the oscillation period and amplitude depending on the parameters of the model. In contrast, in the deformed thermal-AdS geometry, the complexity grows exponentially at late times and does not display oscillatory behavior. We then investigate string dynamics in the string frame and find that the Krylov complexity exhibits oscillatory behavior in both confining backgrounds. This behavior is traced to the emergence of an additional turning point in the deep infrared of the string-frame geometry. Motivated by this observation, we further analyze massive-particle dynamics in the string frame, where we find qualitatively similar behavior in the two confining backgrounds. Our results demonstrate that the Krylov complexity of confined holographic systems depends sensitively on the choice of probe and frame, while also revealing a closer correspondence between the two confining geometries in the string-frame description.
We perform a systematic holographic study of Krylov complexity for a wide class of confining quantum field theories. Using the geometric prescription that identifies the time derivative of the complexity with the proper momentum of a massive probe, we analyse radial geodesics in several top-down gravity duals exhibiting confinement and a mass gap. In all geometries with a smooth infrared end-of-space we uncover a robust and universal qualitative feature: Krylov complexity exhibits oscillatory behaviour. The oscillation frequency is controlled by the confinement scale, while the amplitude depends on both the ultraviolet cutoff and the infrared scale. Additional conserved charges modify these patterns without altering their qualitative structure. We further compare our results with the Krylov complexity of the longitudinally perturbed Ising model. The qualitative agreement suggests that oscillatory behaviour of Krylov complexity constitutes a universal signature of confinement and provides a sensitive probe of infrared reorganisation in strongly coupled quantum field theories.
Following the holographic proposal that identifies the growth rate of Krylov complexity with the proper radial momentum of an infalling massive probe, we study Krylov complexity in Lifshitz and hyperscaling-violating backgrounds. For pure Lifshitz geometries, we find quadratic complexity growth for all values of the dynamical exponent. For hyperscaling-violating backgrounds, we show that the hyperscaling-violating exponent directly controls the late-time growth exponent, with a special limiting case exhibiting oscillatory behavior with a logarithmic envelope. We also extend our analysis to top-down string constructions, confirming the robustness of the scaling behaviors obtained in the bottom-up approach. Our analysis establishes that the momentum-Krylov correspondence extends naturally to non-relativistic holographic settings and remains well-defined despite the causal pathologies of Lifshitz spacetimes.
Krylov Subspace Dynamics as Near-Horizon AdS2_22​ Holography
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 AdS2_22​ 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 AdS2_22​ throat. This correspondence identifies the linear growth rate of Lanczos coefficients with the Hawking temperature, α=πTα=πTα=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,R)SL(2, \mathbb{R})SL(2,R) representation, revealing that the emergent geometry of Krylov subspace is a reflection of the near-horizon AdS spacetime.
We investigate the growth of the black hole interior in three-dimensional gravity from the boundary theory. For the two-sided BTZ black hole, we propose a boundary reconstruction of the time dependence of a codimension-one surface in terms of correlation functions of smeared operators in the thermofield double state, reproducing the characteristic late-time linear growth predicted by the complexity-volume proposal. Using the Chern--Simons formulation of three-dimensional gravity, these nonlocal correlators are represented by bulk Wilson lines with smeared endpoints, extending the familiar connection between Wilson lines and codimension-two observables underlying holographic entanglement entropy to codimension-one observables. We then ask whether the same geometric growth is captured by Krylov complexity. For the smeared operators, we extract the Lanczos data from their correlation functions and find that operator Krylov complexity reproduces the late-time linear growth of the black hole interior, extending previous connections between operator growth and bulk geometry to AdS3_33​. By contrast, the Krylov spread complexity of the thermofield double state, obtained from the semiclassical gravitational partition function, does not exhibit the linear growth of the bulk volume within the regime accessible to our analysis. Our results therefore point to a distinguished role for operator Krylov complexity in encoding black hole interior growth beyond two-dimensional gravity, while highlighting a qualitative distinction between operator and state notions of Krylov complexity.
This thesis investigates quantum complexity in the AdS/CFT duality. We study a recently proposed measure of quantum complexity, the spread complexity. This is a measure of the number of orthonormal basis vectors needed to describe a given quantum state (its “spread” in the Hilbert space), naturally selecting a preferred basis adapted to the dynamics of the state which minimises this quantity. This thesis demonstrates that the spread complexity provides a quantum information- theoretic quantity which directly connects to bulk dynamics in holography. Quantum complexity has long been studied in the context of holography. One result of these investigations is a conjecture relating the rate of change of the complexity of certain states in the CFT to the radial momentum of a dual massive particle falling along a time-like geodesic in the bulk spacetime. We demonstrate that there is an exact match between the rate of change of the spread complexity of an excited 2D CFT state and the dual particle’s radial momentum, measured in appropriate coordinates. As a toy model of the localised bulk particle state, we show that a similar relationship exists between the spread complexity and momentum of coherent states of the quantum harmonic oscillator. The spread complexity of these CFT states can be expressed analytically by relating them to a discrete series representation of SL(2,R). Additionally, we find analytical results for the spread complexity when the excited CFT state is evolved by the modular Hamiltonian of an interval region in the CFT, and by the half-sided modular Hamiltonian associated with a further sub-interval region. Next, we study the spread complexity of CFT states which describe a particle on the BTZ black hole background. We also study the spread complexity of local states in the semi-classical bulk theory. This provides a direct bulk interpretation of the spread complexity to complement our CFT calculations. Finally, we explore the extensivity of the spread complexity of states and of the Krylov complexity of density matrix operators in non-interacting bipartite systems.
The boundary quantity that captures the growth of black-hole interiors quantified by holographic complexity remains unknown beyond 2d dilaton gravity. We provide a critical analysis of a partition-function construction of Krylov spread complexity for thermofield-double states that provides a dimension-independent boundary reconstruction from semiclassical holographic partition functions, while developing the present dynamical and bulk construction for the BTZ saddle. In the double-scaled Sachdev-Ye-Kitaev model, where exact and semiclassical results can be compared, we show that the classical limit is reliable only when taken after the complexity has been reconstructed; taking this limit at the level of individual Lanczos coefficients discards essential information. Applying the construction to a large-central-charge two-dimensional conformal field theory above the Hawking-Page temperature dual to a Bañados-Teitelboim-Zanelli black hole, we find an intermediate departure from early-time quadratic growth followed by behavior compatible with a return toward asymptotically quadratic growth, rather than the linear late-time behavior of the volume and the standard finite-functional complexity = anything class. We then match this boundary behavior to a generalized complexity = anything bulk object built from an infinite series of extrinsic-curvature invariants. The construction provides a systematic route from black-hole thermodynamics to Krylov dynamics and can naturally be extended to higher-dimensional holographic black holes.