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arxiv: 2606.20220 · v2 · pith:7FUTS6BWnew · submitted 2026-06-18 · ✦ hep-th

Higher-loop wormhole length in sine-dilaton gravity from DSSYK Krylov complexity

Eleonora Alfinito , Matteo Beccaria This is my paper

Pith reviewed 2026-06-26 16:02 UTC · model grok-4.3

classification ✦ hep-th
keywords wormhole lengthsine-dilaton gravityDSSYKKrylov complexitysemiclassical expansionLiouville equationshigher-loop correctionscumulants
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The pith

Quantum wormhole length in sine-dilaton gravity equals Krylov spread complexity of the double-scaled SYK model at five-loop order.

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

The paper shows that the quantum wormhole length equals the Krylov spread complexity in the double-scaled SYK model. It computes the five-loop semiclassical expansion of this complexity at infinite temperature by applying singular perturbation to the operator Liouville-type equations of motion. The same technique yields the Krylov variance and third cumulant, which correspond to connected correlation functions of the length operator. Small- and large-time regimes are analyzed, including an all-order resummation of the perturbative series for the large-time linear growth slope together with leading non-perturbative corrections.

Core claim

The length of the quantum wormhole in sine-dilaton gravity is identical to the Krylov spread complexity in the double-scaled SYK model; this identification permits a systematic five-loop semiclassical computation of the complexity via singular perturbation of the associated operator Liouville equations, together with the variance, third cumulant, and an all-order resummation of the large-time slope.

What carries the argument

Singular perturbation applied to the operator Liouville-type equations of motion, which generates the semiclassical loop expansion of the Krylov complexity.

If this is right

  • The five-loop result supplies concrete higher-order corrections to the wormhole length at infinite temperature.
  • The method also determines the connected two- and three-point functions of the length operator via the variance and third cumulant.
  • An all-order resummation exists for the coefficient of linear growth of the wormhole length at late times.
  • Leading non-perturbative corrections to the same large-time slope can be extracted from the same framework.

Where Pith is reading between the lines

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

  • If the identification persists, numerical evaluation of Krylov complexity in finite-N SYK chains could supply quantitative predictions for wormhole lengths that are otherwise difficult to access in gravity.
  • The singular-perturbation technique may extend to other observables that admit a Liouville-type operator description, such as higher moments of the length distribution.
  • The all-order resummation of the slope suggests that the late-time growth rate admits a closed-form expression independent of loop order.

Load-bearing premise

The equality between wormhole length and Krylov spread complexity remains valid at every order in the semiclassical expansion.

What would settle it

An explicit mismatch between the five-loop perturbative terms obtained from the DSSYK equations and the corresponding wormhole-length observables computed directly in sine-dilaton gravity would disprove the identification.

Figures

Figures reproduced from arXiv: 2606.20220 by Eleonora Alfinito, Matteo Beccaria.

Figure 1
Figure 1. Figure 1: Comparison between the numerical evaluation of the Krylov variance and third-order [PITH_FULL_IMAGE:figures/full_fig_p026_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left panel: small time behaviour of the Krylov complexity at J “ 1 and q “ 1 ´ 1{4. The black dashed line is the result of integration of (2.21) with a truncated Krylov basis with 103 states. The red line is the quadratic function 1 1´q pJ tq 2 . The green line is our expansion in (8.1). Right panel: large time behaviour of the Krylov complexity at J “ 1 and q “ 1 ´ 1{4. The black dashed line is the result… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison between the numerically estimated slope [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗
read the original abstract

The quantum wormhole length in sine-dilaton gravity has been shown to equal the Krylov spread complexity in the double-scaled SYK model. In the infinite temperature limit, we compute the five-loop semiclassical expansion of DSSYK complexity by singular perturbation of the operator Liouville-type equations of motion, extending the existing one-loop results. The same method is applied to evaluate the Krylov variance and third-order cumulant, related to the connected two- and three-point functions of the length operator at coincident points. The small- and large-time behaviour of these observables is also studied. In particular, for the large-time slope of the wormhole linear growth, we determine the all-order resummation of the perturbative series, and the leading non-perturbative corrections.

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

0 major / 1 minor

Summary. The paper claims that the quantum wormhole length in sine-dilaton gravity equals the Krylov spread complexity in the double-scaled SYK model. In the infinite temperature limit, it computes the five-loop semiclassical expansion of DSSYK complexity via singular perturbation of the operator Liouville-type equations of motion, extending prior one-loop results. The same method is used to evaluate the Krylov variance and third-order cumulant (related to connected two- and three-point functions of the length operator at coincident points), along with small- and large-time behaviors; an all-order resummation of the perturbative series is given for the large-time slope of the wormhole linear growth, together with leading non-perturbative corrections.

Significance. If the wormhole-length/Krylov-complexity identification persists order-by-order, the manuscript supplies a concrete higher-order verification and extension in a solvable holographic model. Credit is due for the five-loop computation, the application of singular perturbation to operator equations, the extraction of higher cumulants, and especially the all-order resummation of the linear-growth slope together with non-perturbative corrections; these constitute technical advances that furnish explicit data for testing the correspondence beyond one loop.

minor comments (1)
  1. The relation between the length operator cumulants and the connected correlation functions is stated in the abstract but would benefit from an explicit equation or short derivation in the main text (e.g., near the definition of the cumulants) to make the mapping fully transparent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript, their recognition of the technical contributions including the five-loop computation and all-order resummation, and their recommendation to accept. There are no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No circularity; new perturbative computation extends external identification

full rationale

The paper cites the wormhole length = Krylov complexity equality as previously shown in the literature and performs an independent five-loop computation of DSSYK complexity via singular perturbation of the operator Liouville equations, extending one-loop results. No step reduces by construction to a fitted input, self-definition, or load-bearing self-citation chain; the higher-order terms and resummations are derived from the perturbation method itself. The assumption that the identification holds order-by-order is an external modeling choice, not an internal circular reduction. The derivation chain for the new observables is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities can be extracted. The mapping between gravity and SYK quantities is presupposed from earlier work.

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Reference graph

Works this paper leans on

39 extracted references · 15 linked inside Pith

  1. [1]

    Sachdev and J

    S. Sachdev and J. Ye,Gapless Spin Fluid Ground State in a Random, Quantum Heisenberg Magnet,Phys. Rev. Lett.70(1993) 3339 [cond-mat/9212030]

    Pith/arXiv arXiv 1993

  2. [2]

    Sachdev,Holographic Metals and the Fractionalized Fermi Liquid,Phys

    S. Sachdev,Holographic Metals and the Fractionalized Fermi Liquid,Phys. Rev. Lett.105 (2010) 151602 [1006.3794]

    Pith/arXiv arXiv 2010

  3. [3]

    Maldacena and D

    J. Maldacena and D. Stanford,Remarks on the Sachdev-Ye-Kitaev Model,Phys. Rev. D94 (2016) 106002 [1604.07818]

    Pith/arXiv arXiv 2016

  4. [4]

    J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S. H. Shenker et al.,Black Holes and Random Matrices,JHEP05(2017) 118 [1611.04650]

    Pith/arXiv arXiv 2017

  5. [5]

    Maldacena, S

    J. Maldacena, S. H. Shenker and D. Stanford,A Bound on Chaos,JHEP08(2016) 106 [1503.01409]

    Pith/arXiv arXiv 2016

  6. [6]

    Maldacena, D

    J. Maldacena, D. Stanford and Z. Yang,Conformal Symmetry and Its Breaking in Two Dimensional Nearly Anti-De-Sitter Space,PTEP2016(2016) 12C104 [1606.01857]. K. Jensen,Chaos in AdS2 Holography,Phys. Rev. Lett.117(2016) 111601 [1605.06098]. G. Sárosi,AdS 2 holography and the SYK model,PoSModave2017(2018) 001 [1711.08482]

    Pith/arXiv arXiv 2016

  7. [7]

    Berkooz, M

    M. Berkooz, M. Isachenkov, V. Narovlansky and G. Torrents,Towards a Full Solution of the LargeNDouble-Scaled Syk Model,JHEP03(2019) 079 [1811.02584]

    Pith/arXiv arXiv 2019

  8. [8]

    T. G. Mertens, G. J. Turiaci and H. L. Verlinde,Solving the Schwarzian via the Conformal Bootstrap,JHEP08(2017) 136 [1705.08408]

    Pith/arXiv arXiv 2017

  9. [9]

    Engelsöy, T

    J. Engelsöy, T. G. Mertens and H. Verlinde,An investigation of AdS2 backreaction and holography,JHEP07(2016) 139 [1606.03438]. T. G. Mertens and G. J. Turiaci,Solvable models of quantum black holes: a review on Jackiw–Teitelboim gravity,Living Rev. Rel.26(2023) 4 [2210.10846]

    Pith/arXiv arXiv 2016

  10. [10]

    Berkooz, P

    M. Berkooz, P. Narayan and J. Simon,Chord Diagrams, Exact Correlators in Spin Glasses and Black Hole Bulk Reconstruction,JHEP08(2018) 192 [1806.04380]

    Pith/arXiv arXiv 2018

  11. [11]

    Berkooz and O

    M. Berkooz and O. Mamroud,A Cordial Introduction to Double Scaled Syk,Rept. Prog. Phys. 88(2025) 036001 [2407.09396]

    arXiv 2025

  12. [12]

    Yang,The Quantum Gravity Dynamics of Near Extremal Black Holes,JHEP05(2019) 205 [1809.08647]

    Z. Yang,The Quantum Gravity Dynamics of Near Extremal Black Holes,JHEP05(2019) 205 [1809.08647]. T. G. Mertens,The Schwarzian theory — origins,JHEP05(2018) 036 [1801.09605]. A. Blommaert, T. G. Mertens and H. Verschelde,The Schwarzian Theory - a Wilson Line Perspective,JHEP12(2018) 022 [1806.07765]. L. V. Iliesiu, S. S. Pufu, H. Verlinde and Y. Wang,An Ex...

    arXiv 2019

  13. [13]

    Blommaert, T

    A. Blommaert, T. G. Mertens and S. Yao,Dynamical Actions and Q-Representation Theory for Double-Scaled Syk,JHEP02(2024) 067 [2306.00941]. 33

    arXiv 2024

  14. [14]

    Blommaert, T

    A. Blommaert, T. G. Mertens and J. Papalini,The Dilaton Gravity Hologram of Double-Scaled Syk,JHEP06(2025) 050 [2404.03535]

    arXiv 2025

  15. [15]

    Blommaert, A

    A. Blommaert, A. Levine, T. G. Mertens, J. Papalini and K. Parmentier,An Entropic Puzzle in Periodic Dilaton Gravity and Dssyk,JHEP07(2025) 093 [2411.16922]

    arXiv 2025

  16. [16]

    H. W. Lin,The Bulk Hilbert Space of Double Scaled SYK,JHEP11(2022) 060 [2208.07032]

    arXiv 2022

  17. [17]

    Susskind,Computational Complexity and Black Hole Horizons,Fortsch

    L. Susskind,Computational Complexity and Black Hole Horizons,Fortsch. Phys.64(2016) 24 [1403.5695]. D. Stanford and L. Susskind,Complexity and Shock Wave Geometries,Phys. Rev. D90 (2014) 126007 [1406.2678]. A. R. Brown, D. A. Roberts, L. Susskind, B. Swingle and Y. Zhao,Complexity, Action, and Black Holes,Phys. Rev. D93(2016) 086006 [1512.04993]. A. Belin...

    Pith/arXiv arXiv 2016

  18. [18]

    Rabinovici, A

    E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner,A bulk manifestation of Krylov complexity,JHEP08(2023) 213 [2305.04355]

    arXiv 2023

  19. [19]

    Xu,On Chord Dynamics and Complexity Growth in Double-Scaled SYK,JHEP06(2025) 259 [2411.04251]

    J. Xu,On Chord Dynamics and Complexity Growth in Double-Scaled SYK,JHEP06(2025) 259 [2411.04251]

    arXiv 2025

  20. [20]

    Ambrosini, E

    M. Ambrosini, E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner,Operator K-complexity in DSSYK: Krylov complexity equals bulk length,JHEP08(2025) 059 [2412.15318]

    arXiv 2025

  21. [21]

    Nandy, A

    P. Nandy, A. S. Matsoukas-Roubeas, P. Martínez-Azcona, A. Dymarsky and A. del Campo, Quantum dynamics in Krylov space: Methods and applications,Phys. Rept.1125-1128 (2025) 1 [2405.09628]. S. Baiguera, V. Balasubramanian, P. Caputa, S. Chapman, J. Haferkamp, M. P. Heller et al., Quantum Complexity in Gravity, Quantum Field Theory, and Quantum Information S...

    Pith/arXiv arXiv 2025

  22. [22]

    J. M. Maldacena,Eternal Black Holes in Anti-de Sitter,JHEP04(2003) 021 [hep-th/0106112]

    Pith/arXiv arXiv 2003

  23. [23]

    S. E. Aguilar-Gutierrez,Deforming the Double-Scaled SYK & Reaching the Stretched Horizon From Finite Cutoff Holography,2602.06113. S. E. Aguilar-Gutierrez, T. Kukolj and J. Seitz,Q-Askey Deformations of Double-Scaled Syk, 2605.13956

    Pith/arXiv arXiv

  24. [24]

    Harlow and D

    D. Harlow and D. Jafferis,The Factorization Problem in Jackiw-Teitelboim Gravity,JHEP 02(2020) 177 [1804.01081]. D. Harlow and J.-q. Wu,Algebra of Diffeomorphism-Invariant Observables in Jackiw-Teitelboim Gravity,JHEP05(2022) 097 [2108.04841]. 34 A. R. Brown, H. Gharibyan, H. W. Lin, L. Susskind, L. Thorlacius and Y. Zhao,Complexity of Jackiw-Teitelboim g...

    arXiv 2020

  25. [25]

    M. P. Heller, J. Papalini and T. Schuhmann,Krylov Spread Complexity as Holographic Complexity Beyond Jackiw-Teitelboim Gravity,Phys. Rev. Lett.135(2025) 151602 [2412.17785]

    arXiv 2025

  26. [26]

    M. P. Heller, F. Ori, J. Papalini, T. Schuhmann and M.-T. Wang,De Sitter Holographic Complexity from Krylov Complexity in Dssyk,2510.13986

    arXiv

  27. [27]

    H. W. Lin and D. Stanford,A Symmetry Algebra in Double-Scaled Syk,SciPost Phys.15 (2023) 234 [2307.15725]

    arXiv 2023

  28. [28]

    L. V. Iliesiu, M. Mezei and G. Sárosi,The volume of the black hole interior at late times, JHEP07(2022) 073 [2107.06286]

    arXiv 2022

  29. [29]

    A. Goel, V. Narovlansky and H. Verlinde,Semiclassical Geometry in Double-Scaled Syk, JHEP11(2023) 093 [2301.05732]

    arXiv 2023

  30. [30]

    Okuyama and K

    K. Okuyama and K. Suzuki,Correlators of Double Scaled Syk at One-Loop,JHEP05(2023) 117 [2303.07552]

    arXiv 2023

  31. [31]

    Bossi, L

    L. Bossi, L. Griguolo, J. Papalini, L. Russo and D. Seminara,Sine-Dilaton Gravity Vs Double-Scaled Syk: Exploring One-Loop Quantum Corrections,JHEP06(2025) 152 [2411.15957]

    arXiv 2025

  32. [32]

    Griguolo, J

    L. Griguolo, J. Papalini, L. Russo and D. Seminara,A New Perspective on Dilaton Gravity at Finite Cutoff,2512.21774

    arXiv

  33. [33]

    Bhattacharjee, X

    B. Bhattacharjee, X. Cao, P. Nandy and T. Pathak,Operator Growth in Open Quantum Systems: Lessons from the Dissipative SYK,JHEP03(2023) 054 [2212.06180]. Y. Fu, K.-Y. Kim, K. Pal and K. Pal,Statistics and Complexity of Wavefunction Spreading in Quantum Dynamical Systems,JHEP06(2025) 139 [2411.09390]. H. A. Camargo, Y. Fu, V. Jahnke, K.-Y. Kim and K. Pal,H...

    arXiv 2023

  34. [34]

    Bhattacharjee, P

    B. Bhattacharjee, P. Nandy and T. Pathak,Krylov Complexity in Large Q and Double-Scaled SYK Model,JHEP08(2023) 099 [2210.02474]. Y. Fu, H.-S. Jeong, K.-Y. Kim and J. F. Pedraza,Toward Krylov-Based Holography in Double-Scaled Syk,JHEP05(2026) 056 [2510.22658]. A. Grabarits and A. del Campo,Universal Growth of Krylov Complexity Across a Quantum Phase Transi...

    arXiv 2023

  35. [35]

    Nandy,Tridiagonal Hamiltonians Modeling the Density of States of the Double-Scaled Syk Model,JHEP01(2025) 072 [2410.07847]

    P. Nandy,Tridiagonal Hamiltonians Modeling the Density of States of the Double-Scaled Syk Model,JHEP01(2025) 072 [2410.07847]. V. Balasubramanian, J. M. Magan, P. Nandi and Q. Wu,Spread Complexity and the Saturation of Wormhole Size,Phys. Rev. D113(2026) 046004 [2412.02038]. 35

    arXiv 2025

  36. [36]

    Erdős and D

    L. Erdős and D. Schröder,Phase Transition in the Density of States of Quantum Spin Glasses,Math. Phys. Anal. Geom.17(2014) 441 [1407.1552]

    Pith/arXiv arXiv 2014

  37. [37]

    D. E. Parker, X. Cao, A. Avdoshkin, T. Scaffidi and E. Altman,A Universal Operator Growth Hypothesis,Phys. Rev. X9(2019) 041017 [1812.08657]

    arXiv 2019

  38. [38]

    Okuyama,Non-Perturbative Corrections in the Semi-Classical Limit of Double-Scaled Syk, JHEP06(2025) 044 [2501.15501]

    K. Okuyama,Non-Perturbative Corrections in the Semi-Classical Limit of Double-Scaled Syk, JHEP06(2025) 044 [2501.15501]

    arXiv 2025

  39. [39]

    J. L. F. Barbón, E. Rabinovici, R. Shir and R. Sinha,On The Evolution Of Operator Complexity Beyond Scrambling,JHEP10(2019) 264 [1907.05393]. 36

    arXiv 2019