Pith. sign in

REVIEW 3 major objections 4 minor 49 references

$R^2$ corrections to Complexity Growth with a Probe String

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read For a stationary probe string in a Gauss-Bonnet AdS black brane, the complexity growth rate is exactly $T_s L^2 \pi T$ and independent of the Gauss-Bonnet coupling at fixed temperature.

desk verdict A clean, routine probe-string calculation whose static result is a kinematic identity — the moving-string λ_GB suppression is the real new content, but the CA dictionary remains an external assumption. read the letter →

arxiv 2507.20841 v2 pith:PTPMOVVX submitted 2025-07-28 hep-ph

Wen-Bin Chang This is my paper
classification hep-ph
keywords holographiccomplexityequalsactionGauss-BonnetgravityR^2correctionsNambu-Gotoprobestringblackbranegrowth
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper aims to extend the Complexity=Action dictionary to include the leading $R^2$ stringy corrections, modeled by Gauss-Bonnet gravity. It adds a probe string to the bulk and computes the time derivative of its Nambu-Goto action over the Wheeler-DeWitt patch, treating that derivative as the holographic complexity growth. The central result is that a stationary string gives exactly $dS_{NG}/dt = T_s L^2 \pi T$, with no dependence on the Gauss-Bonnet coupling $\lambda_{GB}$ at fixed temperature, while moving strings grow more slowly, with the suppression stronger for larger $\lambda_{GB}$. In all cases the growth rate is linear in temperature, so hotter thermal states complexify faster. These results matter because higher-derivative corrections are the leading quantum and stringy corrections to gravity, and the paper argues they leave the static growth formula untouched while softening the growth of moving nonlocal probes.

What carries the argument

The central object is the time derivative of the Nambu-Goto action of a probe worldsheet, $dS_{NG}/dt = T_s \int_{z_h}^{\infty} dz \, \sqrt{-g_{\mathrm{ind}}(z)}$, integrated over the Wheeler-DeWitt patch and identified with complexity growth. For a string moving with velocity $v$, the worldsheet is parameterized by $t=\tau$, $r=\sigma$, $\phi = v\tau + \xi(\sigma)$, and the machinery is the integral (8), which after eliminating $\xi'$ via the conserved momentum $\Pi_\xi$ and the turning-point conditions $a f_{GB}(z_c) = v^2$ and $a f_{GB}(z_c) L^4 = \Pi_\xi^2 z_c^4$ becomes the compact velocity- and coupling-dependent expression (13). The static limit is evaluated analytically using $z_h = \sqrt{a}/(\pi T)$ to yield the $\lambda_{GB}$-independent rate $T_s L^2 \pi T$.

What would settle it

Compute the full gravitational action of the Wheeler-DeWitt patch for the same Gauss-Bonnet black brane, including its boundary terms, and check whether the growth rate at fixed temperature is independent of $\lambda_{GB}$; any $\lambda_{GB}$ dependence would mean the probe-string action is not tracking the Complexity=Action complexity of the state.

Watch

Extended reading notes

Core claim

Under the probe-string version of the Complexity=Action conjecture, the growth of holographic complexity for a fundamental string in a Gauss-Bonnet AdS5 black brane is governed by the velocity-dependent integral in Eq. (13). The paper's central finding is the exact static result: for $v=0$, the integral collapses to $T_s L^2 \sqrt{a}/z_h = T_s L^2 \pi T$, independent of $\lambda_{GB}$. For nonzero velocity, the growth is symmetric in $v$ and monotonically decreasing in $|v|$, and increasing the Gauss-Bonnet coupling further suppresses the moving-string growth while the overall linear temperature dependence is retained. The paper concludes that $R^2$ corrections leave the stationary probe unchanged and reduce the moving probe's complexity growth.

Load-bearing premise

The results stand on treating the time derivative of the probe string's Nambu-Goto action, integrated over the Wheeler-DeWitt patch, as the holographic dual of complexity growth; if complexity really equals the full gravitational action of that patch, the probe-action calculation does not measure it.

Editorial extensions

If this is right

  • A stationary probe string has exact complexity growth $T_s L^2 \pi T$ in the Gauss-Bonnet background, unmodified by the coupling at fixed temperature.
  • Probe-string motion always suppresses complexity growth, with the suppression depending only on $|v|$ because the action is symmetric under $v \to -v$.
  • Stronger $R^2$ corrections reduce the growth rate of moving strings, so the main effect of higher-derivative corrections is to slow nonlocal probe complexification.
  • Linear temperature scaling persists for every velocity and coupling considered, meaning hotter black branes always drive faster complexity growth.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the same probe dictionary were applied to other curvature-squared terms beyond the special Gauss-Bonnet combination, the static protection might fail, since it relies on the specific scaling $z_h = \sqrt{a}/(\pi T)$; a direct computation with $R_{\mu\nu}^2$ or $R^2$ alone would test this.
  • The velocity-suppression pattern resembles friction and drag phenomena for moving probes in holographic plasmas, so comparing the slope reduction with drag-force and jet-quenching parameters could clarify whether the effect is dissipative or purely geometric.
  • Because the derivation uses a probe, it concerns a nonlocal observable rather than the full boundary state; testing whether the $\lambda_{GB}$ independence of the static rate persists for other probe shapes would show how universal the protection is.
  • Extending the calculation to $R^4$ corrections, as the paper suggests, would reveal whether the static rate is protected against all higher-derivative corrections or only against the leading $R^2$ term.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript studies the effect of Gauss-Bonnet (GB) corrections on holographic complexity growth in a five-dimensional AdS black brane. Following Nagasaki's probe-string approach, the authors introduce a fundamental string with worldsheet ansatz (7), compute the time derivative of its Nambu-Goto action, and interpret this quantity as the complexity growth under the Complexity=Action conjecture. The central results are that the v=0 growth rate equals T_s L^2 pi T independent of the GB coupling at fixed temperature, that moving strings have smaller growth rates with the suppression increasing with |v| and lambda_GB, and that all growth rates are linear in the temperature. The derivation from Eq. (8) to Eq. (13) is algebraic and checkable, and the v=0 formula (14) and the linear-T scaling follow directly from the temperature relation (6).

Significance. If the probe-string dictionary is accepted, the manuscript provides a simple, self-contained example of how higher-curvature corrections modify a probe observable that has been proposed as a holographic complexity measure. Its strengths are that it uses no fitted parameters, states the action and background explicitly, gives an exact analytic result at v=0, and shows the linear temperature scaling by a clean rescaling argument. However, the physical interpretation is entirely dependent on an assumed identification between the time derivative of the Nambu-Goto action of a probe string and CA complexity growth; this dictionary is not derived from the standard CA conjecture, which uses the full gravitational action of the Wheeler-DeWitt patch. Consequently, the results are best regarded as properties of a probe-string observable, and their status as statements about complexity depends on an external assumption.

major comments (3)
  1. [Sec. III, first paragraph and Eq. (8)] The identification of dS_NG/dt with complexity growth under the CA conjecture is assumed from Nagasaki [24] rather than derived. The standard CA conjecture equates complexity with the action of the full Wheeler-DeWitt patch in the gravitational theory [12,13], whereas Eq. (8) computes the Nambu-Goto action of a probe string on a single worldsheet. Because this dictionary is the basis for the abstract's and Sec. IV's claims about complexity, it needs to be justified or the results should be explicitly reframed as statements about the growth of a probe-string observable, not about CA complexity.
  2. [Sec. II, Eq. (3), and Sec. III, Eq. (7)] The worldsheet ansatz (7) uses a coordinate phi with phi = v tau + xi(sigma), but the displayed background metric (3) contains no such angular coordinate. If phi is meant to denote one of the flat boundary coordinates x^i, this identification should be stated explicitly; if phi is a genuine angular coordinate, the metric (3) is incomplete and Eq. (8) cannot be obtained from the displayed line element. In addition, the statement that the action is integrated 'over the WDW patch' reduces here to a radial integral from z_h to infinity; the relation between this integration region and the standard WDW patch should be clarified.
  3. [Sec. III, Eq. (14)] The v=0 result dS_NG/dt = T_s L^2 pi T is independent of the GB coupling only because the factor sqrt(a)/z_h equals pi T by the temperature relation (6). This is a kinematic identity forced by the normalization of the temperature, not a dynamical statement about R^2 corrections. The text should avoid presenting Eq. (14) as a nontrivial prediction about GB gravity; it is a consistency check of the calculation, while the actual GB dependence appears only in the numerical results for moving strings.
minor comments (4)
  1. [Sec. II, Eq. (5)] The function fGB(z) is written in a form that is singular at lambda_GB = 0; the plots in Figs. 1-3 include lambda_GB = 0, so the text should state that the lambda_GB -> 0 limit is taken in Eq. (5) before numerical evaluation.
  2. [Sec. III, Fig. 1] The caption and text state that the action growth 'decreases monotonically' with |v|, but the figure covers only the range |v| <= 0.6 and no analytic proof of monotonicity is given; the statement should be softened or the numerical range extended.
  3. [Sec. III, Eq. (13)] The derivation of Eq. (13) from Eqs. (10)-(12) is not shown in detail; in particular, the use of the critical-point conditions (11) and (12) to eliminate xi' and Pi_xi should be explained in a few lines so that the reader can reproduce the integrand.
  4. [Sec. II, Eq. (3)] The background metric is written for the asymptotically AdS planar black brane, but the relation of the coordinate phi in the probe-string ansatz to one of the x^i coordinates is never stated; a short sentence defining phi would remove the ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all results are direct integrations of the stated probe-string action, with no fitted parameters and no load-bearing self-citation; the Nagasaki dictionary is an external assumption, not a circular reduction.

full rationale

The paper does not fit any parameter to data. It specifies the GB-AdS5 background (Eqs. (3)-(6)), defines the probe-string worldsheet (Eq. (7)), and integrates the NG action over the radial range (Eq. (8)); the subsequent equations (9)-(13) are algebraic rearrangements. The v=0 result (14) is obtained by direct integration and use of the known temperature relation T=sqrt(a)/(pi z_h) (Eq. (6)); the lambda_GB independence at fixed T is a consequence of that relation, not a redefinition of the target quantity. The moving-string suppression and linear T dependence are numerical consequences of the same integral. The only external input is the identification of dS_NG/dt with complexity growth, taken from Nagasaki [24] rather than from the standard CA conjecture. That is an assumption about the dictionary, not a circular derivation; whether it is the correct holographic dictionary is a correctness risk, not evidence that the calculation reduces to its own inputs. The manuscript's self-citations ([32], [33]) appear only in literature lists and are not load-bearing. A separate technical concern, that the displayed metric (3) omits the phi coordinate used in ansatz (7), is an internal consistency issue but does not constitute circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters are fitted; lambda_GB, v, and T are physical inputs. The central claim rests on the interpretive step equating NG action growth with complexity, which is the main domain assumption. No new entities are introduced.

assumptions (3)
  • domain assumption The time derivative of the NG action of a probe string represents holographic complexity growth (probe-string CA proposal).
    Sec. III first paragraph, following Nagasaki [24]. This is an interpretive assumption, not a consequence of the standard CA conjecture.
  • domain assumption The Gauss-Bonnet AdS5 black brane metric (3)-(5) with the causality bound lambda_GB <= 9/100 is the correct background.
    Taken from Cai [49] and Brigante et al. [47]; the paper does not derive this solution.
  • domain assumption The probe string worldsheet ansatz (7) (t=tau, r=sigma, phi=v tau+xi(sigma)) captures the nonlocal Wilson line operator.
    Eq. (7) in Sec. III, from the Nagasaki construction [24,25].

how reviews work

0 comments
Cite this review

Pith. "Pith review of $R^2$ corrections to Complexity Growth with a Probe String." pith.science (2026). https://pith.science/paper/PTPMOVVX

@misc{pith2026250720841,
  author       = {Pith},
  title        = {Pith review of: $R^2$ corrections to Complexity Growth with a Probe String},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PTPMOVVX}},
  note         = {Machine review of arXiv:2507.20841}
}
abstract

We investigate the effect of $R^2$ corrections on holographic complexity growth within the framework of the Complexity=Action (CA) conjecture. By introducing a probe string into a Gauss-Bonnet (GB) $AdS$ black brane background, we analyze the time derivative of the Nambu-Goto (NG) action as the holographic dual to complexity growth. Our results indicate that the complexity growth is maximized for a stationary string and is suppressed by its motion. At fixed temperature, the stationary string complexity growth is independent of the GB coupling, whereas that of moving strings is suppressed by stronger $R^2$ corrections. Finally, the growth rate is shown to increase linearly with temperature, confirming that higher temperatures systematically drive the complexity growth.

Figures

Figures reproduced from arXiv: 2507.20841 by the authors.

Figure 1
Figure 1. FIG. 1. Action growth versus string velocity for different [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. shows the same behavior from a complementary viewpoint. The v = 0 result is independent of λGB, while the curves for v = 0.2 and v = 0.4 decrease as λGB increases. Finally, we examine the temperature dependence of the NG action growth in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Action growth versus temperature. Panel (a) shows different string velocities at [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

49 extracted references · 20 canonical work pages

  1. [24]

    Complexity of AdS_5 black holes with a rotating string

    K. Nagasaki, Phys. Rev. D96, 126018 (2017), arXiv:1707.08376 [hep-th]

    work page Pith review arXiv 2017

  2. [1]

    Watrous, (2008), 10.1007/978-0-387-30440-3 428, arXiv:0804.3401 [quant-ph]

    J. Watrous, (2008), 10.1007/978-0-387-30440-3 428, arXiv:0804.3401 [quant-ph]

    arXiv 2008

  3. [2]

    T. J. Osborne, Reports on Progress in Physics75, 022001 (2012)

    work page 2012

  4. [3]

    Swingle, G

    B. Swingle, G. Bentsen, M. Schleier-Smith, and P. Hayden, Phys. Rev. A94, 040302 (2016)

    work page 2016

  5. [4]

    Banerjee, J

    S. Banerjee, J. Erdmenger, and D. Sarkar, JHEP08, 001 (2018), arXiv:1701.02319 [hep-th]

    arXiv 2018

  6. [5]

    Entanglement Entropy from a Holographic Viewpoint

    T. Takayanagi, Class. Quant. Grav.29, 153001 (2012), arXiv:1204.2450 [gr-qc]

    work page Pith review arXiv 2012

  7. [6]

    V. E. Hubeny, M. Rangamani, and T. Takayanagi, JHEP07, 062 (2007), arXiv:0705.0016 [hep-th]

    arXiv 2007

  8. [7]

    Complexity and Boost Symmetry

    Y. Zhao, Phys. Rev. D98, 086011 (2018), arXiv:1702.03957 [hep-th]

    work page Pith review arXiv 2018

Show all 49 references
  1. [8]

    J. A. Cazares, JHEP06, 001 (2013), arXiv:1211.6685 [hep-th]

    2013 arXiv

  2. [9]

    Hayden and J

    P. Hayden and J. Preskill, JHEP09, 120 (2007), arXiv:0708.4025 [hep-th]

    2007 arXiv

  3. [10]

    Susskind, Fortsch

    L. Susskind, Fortsch. Phys.64, 84 (2016), arXiv:1507.02287 [hep-th]

    2016 arXiv

  4. [11]

    Stanford and L

    D. Stanford and L. Susskind, Phys. Rev. D90, 126007 (2014), arXiv:1406.2678 [hep-th]

    2014 arXiv

  5. [12]

    A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y. Zhao, Phys. Rev. Lett.116, 191301 (2016), arXiv:1509.07876 [hep-th]

    2016 arXiv

  6. [13]

    A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle, and Y. Zhao, Phys. Rev. D93, 086006 (2016), arXiv:1512.04993 [hep-th]

    2016 arXiv

  7. [14]

    Cai, S.-M

    R.-G. Cai, S.-M. Ruan, S.-J. Wang, R.-Q. Yang, and R.-H. Peng, JHEP09, 161 (2016), arXiv:1606.08307 [gr-qc]

    2016 arXiv

  8. [15]

    Carmi, S

    D. Carmi, S. Chapman, H. Marrochio, R. C. Myers, and S. Sugishita, JHEP11, 188 (2017), arXiv:1709.10184 [hep-th]

    2017 arXiv

  9. [16]

    Couch, W

    J. Couch, W. Fischler, and P. H. Nguyen, JHEP03, 119 (2017), arXiv:1610.02038 [hep-th]

    2017 arXiv

  10. [17]

    Belin, R

    A. Belin, R. C. Myers, S.-M. Ruan, G. S´ arosi, and A. J. Speranza, Phys. Rev. Lett.128, 081602 (2022), arXiv:2111.02429 [hep-th]

    2022 arXiv

  11. [18]

    Belin, R

    A. Belin, R. C. Myers, S.-M. Ruan, G. S´ arosi, and A. J. Speranza, JHEP01, 154 (2023), arXiv:2210.09647 [hep-th]

    2023 arXiv

  12. [19]

    Jørstad, R

    E. Jørstad, R. C. Myers, and S.-M. Ruan, JHEP07, 223 (2023), arXiv:2304.05453 [hep-th]

    2023 arXiv

  13. [20]

    R. C. Myers and S.-M. Ruan (2024) arXiv:2403.17475 [hep-th]

    2024 arXiv

  14. [21]

    Moosa, JHEP03, 031 (2018), arXiv:1711.02668 [hep-th]

    M. Moosa, JHEP03, 031 (2018), arXiv:1711.02668 [hep-th]. 8

    2018 arXiv

  15. [22]

    F. J. G. Abad, M. Kulaxizi, and A. Parnachev, JHEP01, 127 (2018), arXiv:1705.08424 [hep-th]

    2018 arXiv

  16. [23]

    Z. Fu, A. Maloney, D. Marolf, H. Maxfield, and Z. Wang, JHEP02, 072 (2018), arXiv:1801.01137 [hep-th]

    2018 arXiv

  17. [25]

    Nagasaki, Phys

    K. Nagasaki, Phys. Rev. D98, 126014 (2018), arXiv:1807.01088 [hep-th]

    2018 arXiv

  18. [26]

    An, R.-G

    Y.-S. An, R.-G. Cai, and Y. Peng, Phys. Rev. D98, 106013 (2018), arXiv:1805.07775 [hep-th]

    2018 arXiv

  19. [27]

    Nagasaki, Int

    K. Nagasaki, Int. J. Mod. Phys. A35, 2050152 (2020), arXiv:1912.03567 [hep-th]

    2020 arXiv

  20. [28]

    Nagasaki, Int

    K. Nagasaki, Int. J. Mod. Phys. A38, 2350027 (2023), arXiv:2205.00196 [hep-th]

    2023 arXiv

  21. [29]

    Zhou, X.-M

    Y.-T. Zhou, X.-M. Kuang, and J.-P. Wu, Eur. Phys. J. C81, 768 (2021), arXiv:2104.12998 [hep-th]

    2021 arXiv

  22. [30]

    F. F. Santos, Eur. Phys. J. Plus135, 810 (2020), arXiv:2005.10983 [hep-th]

    2020 arXiv

  23. [31]

    Bravo-Gaete and F

    M. Bravo-Gaete and F. F. Santos, Eur. Phys. J. C82, 101 (2022), arXiv:2010.10942 [hep-th]

    2022 arXiv

  24. [32]

    Chang and D.-f

    W.-B. Chang and D.-f. Hou, Chin. Phys. C48, 034106 (2024)

    2024

  25. [33]

    Complexity Growth in Flavor-Dependent Systems

    W.-B. Chang, X. Chen, and D. Hou, “Complexity Growth in Flavor-Dependent Systems”, 2025, arXiv:2511.22799 [hep-ph]

    2025

  26. [34]

    Guo and Z.-q

    L. Guo and Z.-q. Zhang, Chin. Phys. C49, 035104 (2025)

    2025

  27. [35]

    Guo, S.-W

    W.-D. Guo, S.-W. Wei, Y.-Y. Li, and Y.-X. Liu, Eur. Phys. J. C77, 904 (2017), arXiv:1703.10468 [gr-qc]

    2017 arXiv

  28. [36]

    S. E. Aguilar-Gutierrez, M. P. Heller, and S. Van der Schueren, Phys. Rev. D110, 066009 (2024), arXiv:2305.11280 [hep-th]

    2024 arXiv

  29. [37]

    J. F. Pedraza, A. Russo, A. Svesko, and Z. Weller-Davies, JHEP02, 093 (2022), arXiv:2106.12585 [hep-th]

    2022 arXiv

  30. [38]

    B. Chen, Y. Liu, and B. Yu, JHEP01, 055 (2024), arXiv:2307.15968 [hep-th]

    2024 arXiv

  31. [39]

    S. E. Aguilar-Gutierrez, B. Craps, J. Hernandez, M. Khramtsov, M. Knysh, and A. Shukla, JHEP03, 173 (2024), arXiv:2312.12349 [hep-th]

    2024 arXiv

  32. [40]

    C´ aceres, ´A

    E. C´ aceres, ´A. J. Murcia, A. K. Patra, and J. F. Pedraza, JHEP12, 077 (2024), arXiv:2408.14535 [hep-th]

    2024 arXiv

  33. [41]

    Jiang and Y.-X

    H.-Y. Jiang and Y.-X. Liu, (2025), arXiv:2506.10398 [hep-th]

    2025

  34. [42]

    M. R. Douglas and S. Kachru, Rev. Mod. Phys.79, 733 (2007), arXiv:hep-th/0610102

    2007 arXiv

  35. [43]

    Buchel, R

    A. Buchel, R. C. Myers, and A. Sinha, JHEP03, 084 (2009), arXiv:0812.2521 [hep-th]. 9

    2009 arXiv

  36. [44]

    Aharony, J

    O. Aharony, J. Pawelczyk, S. Theisen, and S. Yankielowicz, Phys. Rev. D60, 066001 (1999), arXiv:hep-th/9901134

    1999 arXiv

  37. [45]

    Aharony and Y

    O. Aharony and Y. Tachikawa, JHEP01, 037 (2008), arXiv:0711.4532 [hep-th]

    2008 arXiv

  38. [46]

    Brigante, H

    M. Brigante, H. Liu, R. C. Myers, S. Shenker, and S. Yaida, Phys. Rev. D77, 126006 (2008), arXiv:0712.0805 [hep-th]

    2008 arXiv

  39. [47]

    Brigante, H

    M. Brigante, H. Liu, R. C. Myers, S. Shenker, and S. Yaida, Phys. Rev. Lett.100, 191601 (2008), arXiv:0802.3318 [hep-th]

    2008 arXiv

  40. [48]

    Zwiebach, Phys

    B. Zwiebach, Phys. Lett. B156, 315 (1985)

    1985

  41. [49]

    Cai, Phys

    R.-G. Cai, Phys. Rev. D65, 084014 (2002), arXiv:hep-th/0109133. 10

    2002 arXiv

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.