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arxiv: 2607.00074 · v1 · pith:A6J7I4K4new · submitted 2026-06-30 · ✦ hep-th

Holographic Spread Complexity from Branes and Strings

Dimitrios Chatzis , Madison Hammond , Carlos Nunez , Alfonso V. Ramallo , Ricardo T. Santamaria This is my paper

Pith reviewed 2026-07-02 18:40 UTC · model grok-4.3

classification ✦ hep-th
keywords Krylov complexityspread complexityholographyD-branesAdS/CFTRouthianABJM theoryfalling particle
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The pith

D-brane and string probes in AdS backgrounds reproduce short-time quadratic growth of Krylov spread complexity when the Routhian enforces fixed charges.

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

The paper applies the falling-particle proposal for spread complexity to concrete string-theory setups with D0-branes in the ABJM dual, D3-branes in AdS5 x S5, and wound fundamental strings. For radial motion the proper-momentum prescription matches the expected quadratic short-time behaviour. When probes carry momentum along isometric directions the naive prescription conflicts with Krylov requirements, so the authors Legendre-transform to the Routhian to obtain the correct fixed-charge description. The monopole two-point function in the boundary theory supplies the survival amplitude that determines the Krylov moments, while radial fluctuations give controlled corrections. The same Routhian treatment works for the rotating D3-brane and clarifies how winding data enter the effective mass of the string probe separately from Noether charges.

Core claim

Embedding the falling-particle picture into top-down holographic models, the Routhian prescription for fixed-charge probes reproduces the expected short-time quadratic behaviour of Krylov spread complexity; this is verified for a D0-brane identified with a dressed monopole in ABJM theory, for a rotating non-BPS D3-brane with centrifugal barrier, and for a wound string that reduces to an effective massive particle.

What carries the argument

The Routhian obtained by Legendre transform of the probe action, which enforces fixed Noether charges and supplies the effective energy whose square root is the proper momentum that measures complexity growth rate.

If this is right

  • Purely radial D0-brane motion reproduces the quadratic short-time growth of spread complexity.
  • The Routhian resolves the apparent conflict with short-time Krylov behaviour when the probe carries isometric momentum.
  • The survival amplitude of the regulated monopole two-point function determines the Krylov moments.
  • Angular momentum on the D3-brane creates a centrifugal barrier that sets a sharp condition for radial infall.
  • Winding data on the fundamental string enter through the effective mass, distinct from Noether charges handled by the Routhian.

Where Pith is reading between the lines

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

  • The Routhian approach could be applied to other extended probes such as Dp-branes in different AdS compactifications to extract parameter dependence.
  • Distinguishing winding numbers from conserved charges may clarify complexity calculations for non-pointlike excitations beyond holography.
  • Including higher-order radial fluctuations offers a controlled way to study quantum corrections to the probe approximation.

Load-bearing premise

The growth rate of spread complexity is measured by the proper momentum of the falling bulk probe.

What would settle it

A direct computation of the short-time expansion of Krylov complexity from the regulated two-point function of the boundary monopole operator that fails to match the quadratic growth predicted by the D0-brane Routhian proper momentum.

read the original abstract

We study Krylov spread complexity in holographic theories using genuine string-theory probes. Building on the proposal that the growth rate of spread complexity is measured by a proper momentum in the bulk, we embed the falling-particle picture in top-down examples. We first analyse a D0 brane in the type IIA AdS$_4\times {\mathbb{CP}}^3$ background dual to ABJM theory, identifying it with a dressed monopole operator in the boundary CFT. For purely radial motion the proper-momentum prescription reproduces the expected quadratic growth of the complexity. When the probe carries momentum along an isometric direction, the naive prescription gives an apparent conflict with the short-time behaviour required of Krylov complexity. We propose that the correct fixed-charge description is obtained by Legendre transforming to the Routhian. We support the D0-brane interpretation through the regulated monopole two-point function, whose survival amplitude determines the Krylov moments, and we show that radial fluctuations give controlled corrections to the effective energy governing the complexity growth. We then extend the analysis to a rotating non-BPS D3 brane in AdS$_5\times S^5$, where angular momentum produces a centrifugal barrier and a sharp condition for radial in-fall. In the falling regime the Routhian prescription again gives the correct short-time behaviour. Finally, we consider a wound fundamental string in AdS$_5\times S^5$, which reduces to an effective massive falling particle. This clarifies the distinction between Noether charges, which require a fixed-charge Routhian treatment, and winding data, which enter through the effective mass. Our results provide a string-theoretic realisation of holographic spread complexity for point-like and extended excitations, making manifest their dependence on field theory parameters.

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 studies Krylov spread complexity in holographic theories via string-theory probes. It analyzes a D0-brane in the type IIA AdS4×CP3 background (dual to ABJM), identified with a dressed monopole operator, showing that the proper-momentum prescription yields the expected short-time quadratic growth for purely radial motion. For probes carrying isometric momentum, an apparent conflict with short-time behavior is resolved by Legendre-transforming to the Routhian at fixed charge. The analysis extends to a rotating non-BPS D3-brane in AdS5×S5 (with centrifugal barrier and radial in-fall condition) and a wound fundamental string in AdS5×S5 (reducing to an effective massive falling particle). The regulated monopole two-point function supports the operator identification, and radial fluctuations are shown to produce controlled corrections to the effective energy.

Significance. If the falling-particle proposal holds, the work supplies top-down string-theoretic realizations of holographic spread complexity for both point-like and extended objects, making explicit the dependence on field-theory parameters including charges and winding numbers. It usefully distinguishes Noether charges (requiring Routhian treatment) from winding data (entering via effective mass) and connects the bulk prescription to a regulated boundary two-point function. These are concrete strengths for a proposal that remains under active development.

major comments (2)
  1. [Abstract and §1] Abstract and §1: The reproduction of short-time quadratic growth via the Routhian rests entirely on the input assumption that spread-complexity growth rate equals proper bulk momentum of a falling particle. No independent computation of the Krylov moments is performed from the regulated two-point function alone; the two-point function serves only for operator identification, rendering the match non-falsifiable within the manuscript.
  2. [D0-brane section] D0-brane section: When isometric momentum is present, the naive proper-momentum prescription conflicts with the required short-time behavior; the Routhian is introduced as a fix, but no derivation is given showing why proper momentum (rather than, e.g., conserved energy) is the correct bulk observable, nor is a boundary-first check supplied to confirm the Routhian restores the correct Krylov moments.
minor comments (2)
  1. [Notation] Clarify the precise definition of the Routhian and its relation to the probe action in the presence of multiple isometric directions.
  2. [D0-brane radial fluctuations] The discussion of radial fluctuations and controlled corrections to the effective energy would benefit from an explicit error estimate or scaling argument.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the positive assessment of its significance. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1: The reproduction of short-time quadratic growth via the Routhian rests entirely on the input assumption that spread-complexity growth rate equals proper bulk momentum of a falling particle. No independent computation of the Krylov moments is performed from the regulated two-point function alone; the two-point function serves only for operator identification, rendering the match non-falsifiable within the manuscript.

    Authors: The work is explicitly framed as an application and extension of the existing proposal that spread-complexity growth is measured by proper bulk momentum. The manuscript's primary contribution is to embed this prescription in concrete top-down string-theory examples (D0-branes in ABJM, rotating D3-branes, and wound strings) while clarifying the role of charges via the Routhian and the distinction between Noether charges and winding data. The regulated two-point function is used solely to support the operator identification of the D0-brane, as stated in the text. We agree that an independent extraction of Krylov moments directly from the boundary correlator would provide a stronger test, but performing such a CFT computation lies outside the scope of the present holographic analysis. The short-time match therefore functions as a consistency check rather than a derivation of the proposal itself. revision: no

  2. Referee: [D0-brane section] D0-brane section: When isometric momentum is present, the naive proper-momentum prescription conflicts with the required short-time behavior; the Routhian is introduced as a fix, but no derivation is given showing why proper momentum (rather than, e.g., conserved energy) is the correct bulk observable, nor is a boundary-first check supplied to confirm the Routhian restores the correct Krylov moments.

    Authors: The preference for proper momentum over conserved energy is inherited from the foundational proposal on which the manuscript builds; the text does not attempt to re-derive this choice. The Routhian is introduced because fixing the Noether charge is required to recover the quadratic short-time growth demanded by Krylov complexity for charged operators; the naive momentum prescription violates this. A direct boundary-first verification that the Routhian reproduces the correct Krylov moments would require constructing the Krylov basis from the charged two-point function in the CFT, which is not performed here. We can add a short clarifying paragraph in §1 that recalls the motivation for the proper-momentum prescription from the prior literature and notes the scope limitation regarding boundary computations. revision: partial

Circularity Check

0 steps flagged

No significant circularity; applies prior falling-particle proposal to new brane setups as consistency checks

full rationale

The derivation applies an external proposal (growth rate equals bulk proper momentum) to D0-brane, D3-brane and wound-string probes, using the Routhian to restore quadratic short-time growth in fixed-charge cases. The reproduction of expected Krylov behaviour is a consistency verification against known field-theory short-time expansion, not a reduction of the output to the input by construction. No fitted parameters are relabelled as predictions, no self-citation chain is shown to be the sole justification for the central claim, and the regulated two-point function is used only to identify the operator, not to close a definitional loop. The work therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract only; main domain assumption is the bulk proper-momentum prescription for complexity growth. No free parameters or invented entities are identifiable from the abstract.

axioms (1)
  • domain assumption Growth rate of spread complexity is measured by proper momentum in the bulk
    Explicitly stated as the proposal the authors build upon.

pith-pipeline@v0.9.1-grok · 5855 in / 1188 out tokens · 33305 ms · 2026-07-02T18:40:27.324307+00:00 · methodology

discussion (0)

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