How does holographic Krylov complexity behave in confining backgrounds for particles and strings?

A radial-momentum shortcut to complexity growth

The paper asks whether confinement leaves a recognizable imprint on Krylov complexity, a measure of how an initially simple operator or state spreads into an increasingly complex combination under time evolution. Rather than building the Krylov basis directly, the authors use a recently proposed holographic correspondence: the growth rate of complexity is taken to be proportional to the proper radial momentum of a probe falling through the bulk spacetime. The probe is either a massive point particle or a string.

This choice sets the whole logic of the study. The complexity curve is read off from the trajectory of the probe, so anything that changes the radial motion, such as a turning point where the probe reverses direction, changes the shape of the complexity curve. The authors work in bottom-up holographic QCD, meaning the gravitational model is built phenomenologically rather than derived from a specific string compactification, and they keep the dual theory in the confined phase throughout.

Two horizonless geometries from one dilaton model

Both backgrounds solve the same five-dimensional Einstein-dilaton action and are asymptotically AdS near the boundary. The deformed AdS-soliton has a capping-off function g(z) that closes the geometry at a finite radial position z0, the soliton tip, and one spatial direction xs is compact with period Ls fixed by regularity at that tip. The deformation is controlled by the parameter a in the form factor A(z) = -a z^2; sending a to zero recovers the standard AdS-soliton.

The deformed thermal-AdS background is obtained by taking the z0 to infinity limit, so g(z) = 1 and the geometry extends smoothly into the deep interior. Its exponential warp factor still modifies the infrared and produces confinement-like features. The paper treats z0 and a as effective phenomenological parameters and scans a broad range of them, noting that the dual theory stays confined for all values considered, with an area-law Wilson loop and a vanishing Polyakov loop.

The two frames matter because different probes couple to different metrics. A point particle is minimally coupled to the Einstein-frame metric, while a fundamental string couples naturally to the string-frame metric, related to the Einstein frame by a dilaton-dependent conformal factor. The paper therefore computes particle dynamics in the Einstein frame, string dynamics in the string frame, and then revisits particles in the string frame.

Particles split into oscillation versus growth

For a massive particle released from rest near the boundary and falling radially in the Einstein frame, the two backgrounds give qualitatively distinct complexity histories. In the deformed AdS-soliton, the trajectory has two turning points: one where g(z0) = 0 at the soliton tip, and one at a finite radial position set by the conserved energy H. The particle bounces between them, and the complexity oscillates periodically, with both the period and the amplitude depending on the model parameters.

In the deformed thermal-AdS background there is no reflecting endpoint, so the radial motion is unbounded and the complexity grows monotonically, with the abstract reporting exponential growth at late times and no oscillatory behavior. The authors emphasize that both geometries describe confined phases, so the difference cannot be attributed to confinement versus deconfinement. Instead, they trace it to the detailed infrared structure: the soliton tip reflects the particle, while the smooth deformed thermal-AdS interior does not.

The paper also notes that the oscillatory particle result in the soliton is qualitatively consistent with earlier top-down confining constructions, which had attributed periodic behavior to a finite Hilbert space. The new element here is that a second, equally confining bottom-up geometry does not reproduce that behavior for particles.

Strings oscillate in both confining backgrounds

Switching the probe to a string and working in the string frame changes the picture. The paper finds oscillatory Krylov complexity in both the deformed AdS-soliton and the deformed thermal-AdS backgrounds, so the sharp particle-level distinction disappears. The authors attribute this to an additional turning point that appears in the string-frame geometry deep in the infrared, which is absent in the corresponding Einstein-frame description.

The mechanism is geometric rather than a new dynamical assumption: the string-frame metric, obtained by applying the dilaton-dependent conformal transformation, develops an effective wall or turning point that stops the probe from propagating arbitrarily deep. In the deformed thermal-AdS case the paper describes this as an effective or imaginary wall beyond which the string worldsheet cannot propagate, dynamically generating an infrared confinement scale. In the soliton case the finite tip plays an analogous role.

This is also where the paper connects to conventional confinement diagnostics. The same infrared structure that produces the linear quark-antiquark potential and the area-law Wilson loop in both backgrounds also produces the extra turning point that makes string complexity oscillate. The authors stress that entanglement measures and quark-antiquark free energy look qualitatively similar across the two geometries, whereas Krylov complexity for particles in the Einstein frame does not.

Frame choice reshapes the particle result

Motivated by the string result, the authors redo the massive-particle analysis in the string frame. They find that particle complexity now behaves qualitatively similarly in the two confining backgrounds, with oscillatory structure in both, substantially modifying the oscillatory-versus-nonoscillatory split seen in the Einstein frame. The string-frame geometry again supplies the extra infrared turning point that drives the oscillation.

In the deformed thermal-AdS string-frame case, the proper momentum rises from zero, peaks, falls to zero at the turning point, reverses sign, and the complexity increases monotonically until that turning point and then decreases monotonically on the outward leg. The paper reports that the time needed to reach the turning point decreases as a increases, so the oscillation frequency rises with the deformation parameter, and that trajectories for different a nearly overlap early and separate only at later times.

The broader lesson the authors draw is that qualitative features of holographic information spreading depend not only on which confining background is chosen but also on the frame in which the probe dynamics is formulated. They note that frame dependence had already appeared in a holographic chaos study of bottom-up QCD models, and they flag it as an open question whether this is an unphysical artifact of bottom-up constructions or would persist in a full top-down setting.

Where the conclusions are bounded

The central computational tool is a conjectured correspondence between complexity growth rate and the proper radial momentum of an infalling probe. The paper states that this prescription is not proven in general, with the analytically controllable AdS3/CFT2 case as the exception, so the complexity curves here inherit the status of that conjecture rather than standing as a first-principles derivation.

The setting is also restricted. The analysis uses the simplest Einstein-scalar framework, two horizonless confining geometries, and treats z0 and a as phenomenological parameters rather than fixing them from a microscopic dual theory. The authors explicitly say it is unclear whether the frame-dependent differences are an unphysical relic of bottom-up constructions. The results therefore describe these two model backgrounds and these probes, not confined holographic systems in general, and they are not statements about real QCD.

Probe and frame, not confinement alone, set the signal

The paper's most decision-relevant point is a comparison, not a single number. If a reader wants to know whether confinement produces oscillatory Krylov complexity, the answer depends on which probe and which frame are used: a massive particle in the Einstein frame oscillates in the deformed AdS-soliton but grows without oscillation in deformed thermal-AdS, while strings in the string frame oscillate in both, and particles in the string frame also oscillate in both.

That pattern makes the infrared turning point the quantity to watch. The soliton tip and the string-frame infrared wall both create a reflecting structure that turns radial motion around, and the complexity signal follows that structure rather than the mere fact of confinement. The paper's own framing supports this priority: it presents the string-frame description as revealing a closer correspondence between the two confining geometries, and it identifies the extra string-frame turning point as the geometric explanation for the difference between particle and string dynamics.

The open question the authors leave is whether this frame sensitivity survives in a top-down holographic QCD setting, and whether the radial-momentum prescription can be tested directly from Lanczos coefficients in the Hilbert space. Until then, the practical reading is that a claim about complexity in a confining background should specify the probe and the frame alongside the geometry, because changing either can flip the qualitative behavior.

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