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arxiv: 2604.07432 · v1 · submitted 2026-04-08 · ✦ hep-th

Recognition: 2 theorem links

· Lean Theorem

Holographic Krylov Complexity for Charged, Composite and Extended Probes

Horatiu Nastase , Carlos Nunez , Dibakar Roychowdhury
Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:58 UTC · model grok-4.3

classification ✦ hep-th
keywords holographic complexityKrylov complexityAdS/CFT correspondenceprobe branesextended operatorsR-chargegiant gravitonsfundamental strings
0
0 comments X

The pith

Holographic Krylov complexity for pointlike probes with internal structure or charge retains the same leading large-time growth as local operators, while genuinely extended probes show distinct subleading and intermediate behavior.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how the holographic version of Krylov complexity, previously studied for local operators, behaves when applied to probes that carry charge, possess composite structure, or are spatially extended. For a charged massive particle and for baryon vertices or giant gravitons, the leading late-time growth follows the linear pattern familiar from conformal field theory operators, with internal motion and charges producing only subleading modifications. For a fundamental string stretched along a spatial direction, the leading growth remains similar but the subleading corrections and the way complexity develops at intermediate times differ in character. This distinction supplies concrete evidence that spread complexity can register the spatial non-locality of an operator. The results therefore enlarge the set of probes for which the holographic dictionary can be tested explicitly and separate universal features from those that depend on the operator's internal or extended nature.

Core claim

For structured yet pointlike bulk probes the leading large-time growth of holographic Krylov complexity retains the linear form characteristic of local operators in conformal theories, while internal structure and induced charges generate informative subleading effects; for a genuinely extended string probe the subleading terms and intermediate-time regimes differ qualitatively, indicating that extended operators support a finer, spatially sensitive notion of spread complexity.

What carries the argument

Holographic dictionary mapping Krylov (spread) complexity of a boundary operator to the dynamics of a bulk probe, applied without modification to charged particles, baryon vertices, giant gravitons, and falling fundamental strings.

If this is right

  • Motion of a probe in the internal S^5 directions supplies a holographic model of symmetry-resolved Krylov complexity.
  • Baryon-vertex and giant-graviton configurations produce subleading corrections that encode their internal charges and structure while preserving the leading growth law.
  • An extended string probe yields subleading and intermediate-time complexity that is sensitive to its spatial extent, unlike the pointlike cases.
  • The distinction between pointlike and extended probes supplies a concrete diagnostic for when operator non-locality modifies complexity growth.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Field-theory definitions of complexity for composite operators may need to incorporate subleading charge-dependent terms that the holographic calculation isolates.
  • The qualitative difference for extended probes suggests that any boundary definition of spread complexity for non-local operators should include a spatial-size parameter absent from local-operator versions.
  • The same probe constructions could be used to compare Krylov complexity with other holographic measures such as entanglement entropy or circuit complexity for the same extended objects.

Load-bearing premise

The holographic dictionary for Krylov complexity that was previously defined for local operators continues to apply without modification to these charged, composite, and extended bulk probes.

What would settle it

An explicit computation of the late-time growth rate for the stretched fundamental string that deviates from the linear-in-time scaling at leading order would falsify the claim that the characteristic conformal growth form survives for extended probes.

read the original abstract

We study the holographic spread/Krylov complexity of operators with non-trivial internal structure and of genuinely extended operators. We first consider a massive particle in AdS$_5\times S^5$ carrying conserved $R$-charge, and show how motion in the internal space modifies the complexity growth, yielding a natural holographic realisation of symmetry-resolved Krylov complexity. We then move to probes that are effectively pointlike from the field-theory viewpoint but possess an intrinsic structure in the bulk: baryon-vertex configurations and giant gravitons. Our results indicate that, for this broad class of structured but pointlike probes, the leading large-time behaviour retains the characteristic form expected for local operators in conformal theories, while the internal structure and induced charges produce informative subleading effects. We also study a genuinely extended probe, a fundamental string falling in AdS while stretched along a spatial direction, as a model for the spread complexity of a non-local operator. In this case, although the leading behaviour still exhibits the expected growth pattern, the subleading terms and intermediate regimes differ qualitatively from those of pointlike probes. This provides concrete evidence that extended operators carry a finer notion of spread complexity, sensitive to their spatial structure. Our results broaden the class of probes for which holographic Krylov complexity can be analysed explicitly, clarify which features are universal and which depend on the nature of the operator, and open a promising route toward a sharper field-theory understanding of complexity for charged, composite and extended excitations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to compute holographic Krylov complexity for a charged massive particle in AdS5×S5, realizing symmetry-resolved complexity via internal-space motion; for baryon-vertex and giant-graviton configurations as structured but pointlike probes, the leading large-time growth retains the form expected for local CFT operators while internal structure and charges produce subleading modifications; and for a stretched fundamental string as an extended probe, the leading growth is similar but subleading terms and intermediate regimes differ qualitatively, indicating that extended operators carry a finer, spatially sensitive notion of spread complexity.

Significance. If the central dictionary extension holds, the work meaningfully broadens holographic Krylov complexity to charged, composite, and extended probes, supplying concrete bulk realizations that distinguish universal leading growth from structure-dependent subleading effects. Explicit treatment of symmetry resolution and the qualitative contrast for the string probe are strengths that could guide field-theory definitions of complexity for non-local operators.

major comments (2)
  1. [Setup and probe definitions (Abstract and early sections)] The central claims rest on applying the late-time geodesic/length functional previously used for local operators directly to the worldline action of the charged particle, the D-brane configurations of baryons/giant gravitons, and the Nambu-Goto action of the stretched string, without a derivation from the CFT operator algebra or a cross-check that reproduces the known linear growth for a local primary. This assumption is load-bearing for both the 'retained leading form' and the 'qualitatively different subleading' statements.
  2. [Extended string probe analysis] For the extended string, the reported qualitative differences in subleading terms and intermediate regimes presuppose that the same dictionary continues to isolate Krylov complexity once spatial extent is present; if the bulk quantity instead mixes with volume or circuit complexity for non-local operators, the interpretation of these differences as statements about Krylov complexity is compromised.
minor comments (2)
  1. The abstract summarizes leading and subleading behaviors without displaying the explicit functional forms, growth rates, or any equations, which hinders immediate assessment of the claimed universality and differences.
  2. Consider adding a brief table or plot comparing the leading coefficients and subleading corrections across the four probe classes to make the distinctions more quantitative and transparent.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We respond point-by-point to the major comments below, indicating where we will revise the presentation to better articulate the scope and assumptions of our holographic approach.

read point-by-point responses

  1. Referee: [Setup and probe definitions (Abstract and early sections)] The central claims rest on applying the late-time geodesic/length functional previously used for local operators directly to the worldline action of the charged particle, the D-brane configurations of baryons/giant gravitons, and the Nambu-Goto action of the stretched string, without a derivation from the CFT operator algebra or a cross-check that reproduces the known linear growth for a local primary. This assumption is load-bearing for both the 'retained leading form' and the 'qualitatively different subleading' statements.

    Authors: We agree that the central results rely on extending the established late-time geodesic dictionary for Krylov complexity to these probes. A first-principles derivation from the CFT operator algebra for charged or composite operators is not provided in this work, as it would require a separate field-theoretic analysis. In the revised manuscript we will expand the introduction and the setup section to state this assumption explicitly, reference the prior literature on local primaries where the dictionary was tested, and include an explicit reduction to the standard linear growth when charges and internal degrees of freedom are switched off. This serves as an internal consistency check. The calculations themselves remain unchanged, but the presentation will be updated to foreground the assumptions. revision: partial

  2. Referee: [Extended string probe analysis] For the extended string, the reported qualitative differences in subleading terms and intermediate regimes presuppose that the same dictionary continues to isolate Krylov complexity once spatial extent is present; if the bulk quantity instead mixes with volume or circuit complexity for non-local operators, the interpretation of these differences as statements about Krylov complexity is compromised.

    Authors: We acknowledge that the interpretation for genuinely extended probes is more tentative. Our bulk computation for the stretched string yields subleading and intermediate-time behavior that differs qualitatively from the pointlike cases. In the revision we will rephrase the relevant discussion and conclusions to present this as a holographic model indicating a spatially sensitive notion of spread complexity, while explicitly noting the possibility of mixing with other complexity measures and the need for future field-theory cross-checks. The reported bulk differences are robust within the calculation, but we will avoid overclaiming a direct field-theory identification. revision: partial

standing simulated objections not resolved
  • A complete derivation of the holographic dictionary from the CFT operator algebra for charged, composite, and extended probes.

Circularity Check

0 steps flagged

No significant circularity: results are explicit bulk computations under an extended dictionary

full rationale

The paper applies the holographic Krylov complexity dictionary (relating late-time growth to bulk probe length or geodesic motion) to new classes of probes by solving the equations of motion for their respective actions—worldline for charged particles, D-brane configurations for baryons and giant gravitons, and Nambu-Goto for the stretched string. The leading linear growth and subleading corrections are read off directly from these solutions in AdS5×S5, without fitting parameters to match prior local-operator results or redefining the complexity measure in terms of the output. No self-citation chain, ansatz smuggling, or uniqueness theorem is invoked to force the outcomes; the calculations are independent of the target behaviors once the dictionary is assumed. This constitutes a standard extension via explicit computation rather than a closed loop.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard AdS/CFT dictionary for operator growth and on the assumption that Krylov complexity can be read off from bulk probe dynamics; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption AdS/CFT correspondence maps bulk probe motion to boundary operator evolution
    Invoked throughout to translate bulk geodesics and string dynamics into field-theory complexity growth.
  • domain assumption Krylov complexity growth for local operators is known and serves as baseline
    Abstract compares new results to the characteristic linear growth expected for local operators in conformal theories.

pith-pipeline@v0.9.0 · 5569 in / 1404 out tokens · 32878 ms · 2026-05-10T17:58:47.100518+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Krylov state complexity for BMN matrix model

    hep-th 2026-05 unverdicted novelty 5.0

    An analytical method is presented to calculate Lanczos coefficients governing Krylov complexity in the reduced pulsating fuzzy sphere version of the BMN matrix model for large and small deformations.

  2. Krylov complexity for Lin-Maldacena geometries and their holographic duals

    hep-th 2026-04 unverdicted novelty 5.0

    In the BMN matrix model and its holographic duals, Krylov basis states and Lanczos coefficients are uniquely fixed by the model's mass parameter.

Reference graph

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