Holographic complexity of the Klebanov-Strassler background

Andrew R. Frey; Michael P. Grehan; Prakriti Singh

arxiv: 2311.18804 · v3 · submitted 2023-11-30 · ✦ hep-th

Holographic complexity of the Klebanov-Strassler background

Andrew R. Frey , Michael P. Grehan , Prakriti Singh This is my paper

Pith reviewed 2026-05-24 05:04 UTC · model grok-4.3

classification ✦ hep-th

keywords holographic complexityKlebanov-Strassler backgroundgauge/gravity dualityconfinement scaleUV cutoff divergencenonconformal theoriesSU(N) x SU(N+M)

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The pith

Holographic complexity in the Klebanov-Strassler geometry exhibits common dependence on the confinement scale across functionals and a distinct UV divergence due to nonconformality.

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

The paper applies several holographic complexity proposals to the gravity dual of the confining Klebanov-Strassler gauge theory. It finds that these proposals display a shared pattern in how complexity relates to the theory's confinement scale. The analysis also shows that the divergence of complexity as the ultraviolet cutoff is removed differs from the simpler behavior seen in anti-de Sitter space because the dual theory lacks conformal symmetry. A sympathetic reader would care because this provides a concrete test case for extending complexity calculations beyond conformal theories to more realistic confining ones.

Core claim

The complexity of the Klebanov-Strassler background shows a common behavior with the confinement scale for several complexity functionals, and its divergence with the UV cutoff is more complicated than in AdS backgrounds because the theory is nonconformal.

What carries the argument

The Klebanov-Strassler geometry as the gravity dual to the confining SU(N)×SU(N+M) gauge theory, on which multiple holographic complexity functionals are evaluated.

If this is right

Complexity depends on gauge theory parameters through a shared relation to the confinement scale.
The UV divergence of complexity requires more involved treatment than in conformal AdS cases.
The results supply a starting point for complexity calculations in general gauge/gravity dualities.
The example may yield new perspectives on open questions within the holographic complexity program.

Where Pith is reading between the lines

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

The same functionals could be evaluated on other confining supergravity backgrounds to test whether the common confinement-scale behavior is generic.
This approach may clarify how complexity encodes confinement in holographic models of strongly coupled gauge theories.
Finite-temperature deformations of the Klebanov-Strassler solution could be used to check whether the observed patterns survive.

Load-bearing premise

The standard holographic complexity functionals remain well-defined and physically meaningful when applied to the nonconformal Klebanov-Strassler geometry without additional regularization or adjustments beyond those stated.

What would settle it

An explicit computation for the Klebanov-Strassler background showing that complexity does not exhibit common scaling with the confinement scale for one or more of the functionals, or that its UV cutoff divergence exactly matches the form found in AdS.

read the original abstract

We study the complexity of the gravity dual to the confining $SU(N)\times SU(N+M)$ Klebanov-Strassler gauge theory, which is an important test case for holographic complexity in higher-dimensional and nonconformal gauge/gravity dualities. We emphasize the dependence of the complexity on parameters of the gauge theory, finding a common behavior with confinement scale for several complexity functionals. We also analyze how the complexity diverges with the UV cut off, which is more complicated than in AdS backgrounds because the theory is nonconformal. Our results may provide new perspectives on questions in the holographic complexity program as well as a starting point for further studies of complexity in general gauge/gravity dualities.

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.

Desk Editor's Note private letter to a colleague

Abstract reports common confinement-scale behavior for complexity in KS geometry and messier UV divergences, but supplies zero calculations or details to evaluate.

read the letter

The abstract's main point is that several holographic complexity proposals applied to the Klebanov-Strassler background all track the confinement scale in the same way, while the UV cutoff dependence turns out more involved than the usual AdS power-law because the dual theory is nonconformal. That observation, if it holds, would be a concrete data point for complexity outside the conformal case. KS is a standard confining example, so checking how complexity behaves there is a logical extension of the existing program. The abstract does a clean job of stating why the geometry matters and what the parameter dependence looks like at a high level. Beyond that, there is not much to credit because nothing else is shown. The soft spots are straightforward and fairly large. No explicit functionals, no regularization scheme, no numerical values, and no comparison to prior KS calculations appear in the abstract. Without those pieces it is impossible to tell whether the claimed common behavior is robust or sensitive to choices in the cutoff or the functional. The nonconformal UV claim is plausible on general grounds, but again rests on an assertion rather than evidence. This work would mainly interest people already tracking holographic complexity proposals and looking for test cases in confining or higher-dimensional duals. A reader wanting actual numbers or derivations would get little from it. I would not bring the current version to a reading group. It does not contain enough technical content to justify sending it to referees; the authors would need to supply the full calculations and checks first.

Referee Report

0 major / 0 minor

Summary. The manuscript studies holographic complexity in the Klebanov-Strassler background dual to the confining SU(N)×SU(N+M) gauge theory. It reports that several complexity functionals display common behavior with the confinement scale and that the UV divergence of complexity is more complicated than in AdS backgrounds owing to the nonconformal character of the theory.

Significance. If substantiated, the results would extend holographic complexity calculations to a key nonconformal confining background, supplying a concrete test case beyond AdS and potentially informing broader questions in the complexity program. The abstract, however, supplies no explicit functionals, regularization details, numerical outputs, or derivations, so the actual significance cannot be assessed from the provided text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review of our manuscript. The full text provides the details referenced in the abstract, and we address the assessment of significance below.

read point-by-point responses

Referee: The abstract, however, supplies no explicit functionals, regularization details, numerical outputs, or derivations, so the actual significance cannot be assessed from the provided text.

Authors: Abstracts are necessarily brief. The complete manuscript specifies the complexity functionals (with explicit expressions and comparisons across them), the regularization procedure adapted to the nonconformal UV structure, numerical results demonstrating the common scaling with the confinement scale, and the step-by-step derivations of the divergence structure. These elements substantiate the extension of holographic complexity to the Klebanov-Strassler background. revision: no

Circularity Check

0 steps flagged

No circularity detected from available text

full rationale

The abstract describes results on complexity functionals in the Klebanov-Strassler geometry and their relation to confinement scale and UV divergence but supplies no equations, parameter fittings, derivations, or citations. No load-bearing steps are present that could reduce by construction to inputs, self-definitions, or self-citations. The work is therefore treated as self-contained with no evidence of circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; ledger left empty.

pith-pipeline@v0.9.0 · 5618 in / 1069 out tokens · 19795 ms · 2026-05-24T05:04:21.220647+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages · 24 internal anchors

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S. Chapman, H. Marrochio and R.C. Myers, Complexity of Formation in Holography , JHEP 01 (2017) 062 [ 1610.08063]

work page internal anchor Pith review Pith/arXiv arXiv 2017
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Symmetry Breaking and Axionic Strings in the Warped Deformed Conifold

S.S. Gubser, C.P. Herzog and I.R. Klebanov, Symmetry breaking and axionic strings in the warped deformed conifold, JHEP 09 (2004) 036 [ hep-th/0405282]

work page internal anchor Pith review Pith/arXiv arXiv 2004
[29]

The Baryonic Branch of Klebanov-Strassler Solution: a Supersymmetric Family of SU(3) Structure Backgrounds

A. Butti, M. Grana, R. Minasian, M. Petrini and A. Zaffaroni, The Baryonic branch of Klebanov-Strassler solution: A supersymmetric family of SU(3) structure backgrounds , JHEP 03 (2005) 069 [ hep-th/0412187]

work page internal anchor Pith review Pith/arXiv arXiv 2005
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Remarks on the Warped Deformed Conifold

C.P. Herzog, I.R. Klebanov and P. Ouyang, Remarks on the warped deformed conifold , in Modern Trends in String Theory: 2nd Lisbon School on g Theory Superstrings , 8, 2001 [hep-th/0108101]

work page internal anchor Pith review Pith/arXiv arXiv 2001
[31]

Gravity Duals of Supersymmetric SU(N) x SU(N+M) Gauge Theories

I.R. Klebanov and A.A. Tseytlin, Gravity duals of supersymmetric SU(N) x SU(N+M) gauge theories, Nucl. Phys. B 578 (2000) 123 [ hep-th/0002159]

work page internal anchor Pith review Pith/arXiv arXiv 2000
[32]

Complex geometry of conifolds and 5-brane wrapped on 2-sphere

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work page internal anchor Pith review Pith/arXiv arXiv 2001
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Tricomi, Asymptotische Eigenschaften der unvollst¨ andigen Gammafunktion, Math

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Tricomi, Sulla funzione gamma incompleta , Ann

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B¨ ohmer, Paul Eugen,Differenzengleichungen und bestimmte Integrale, Koehler, Liepzig (1939)

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Gravitational action with null boundaries

L. Lehner, R.C. Myers, E. Poisson and R.D. Sorkin, Gravitational action with null boundaries , Phys. Rev. D 94 (2016) 084046 [ 1609.00207]

work page internal anchor Pith review Pith/arXiv arXiv 2016
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Jørstad, R.C

E. Jørstad, R.C. Myers and S.-M. Ruan, Holographic complexity in dS d+1, JHEP 05 (2022) 119 [2202.10684]

work page arXiv 2022
[40]

de Sitter Vacua in String Theory

S. Kachru, R. Kallosh, A.D. Linde and S.P. Trivedi, De Sitter vacua in string theory , Phys. Rev. D 68 (2003) 046005 [ hep-th/0301240]. – 24 –

work page internal anchor Pith review Pith/arXiv arXiv 2003

[1] [1]

Chapman and G

S. Chapman and G. Policastro, Quantum computational complexity from quantum information to black holes and back , Eur. Phys. J. C 82 (2022) 128 [ 2110.14672]. 6We thank C. Nu˜ nez for providing this formula. – 22 –

work page arXiv 2022

[2] [2]

A geometric approach to quantum circuit lower bounds

M.A. Nielsen, A geometric approach to quantum circuit lower bounds , Quant. Inf. Comput. 6 (2006) 213 [ quant-ph/0502070]

work page internal anchor Pith review Pith/arXiv arXiv 2006

[3] [3]

Quantum Computation as Geometry

M.A. Nielsen, M.R. Dowling, M. Gu and A.C. Doherty, Quantum Computation as Geometry , Science 311 (2006) 1133 [ quant-ph/0603161]

work page internal anchor Pith review Pith/arXiv arXiv 2006

[4] [4]

The geometry of quantum computation

M.R. Dowling and M.A. Nielsen, The geometry of quantum computation , Quant. Inf. Comput. 8 (2008) 0861 [ quant-ph/0701004]

work page internal anchor Pith review Pith/arXiv arXiv 2008

[5] [5]

Circuit complexity in quantum field theory

R. Jefferson and R.C. Myers, Circuit complexity in quantum field theory , JHEP 10 (2017) 107 [1707.08570]

work page internal anchor Pith review Pith/arXiv arXiv 2017

[6] [6]

Towards Complexity for Quantum Field Theory States

S. Chapman, M.P. Heller, H. Marrochio and F. Pastawski, Toward a Definition of Complexity for Quantum Field Theory States , Phys. Rev. Lett. 120 (2018) 121602 [ 1707.08582]

work page internal anchor Pith review Pith/arXiv arXiv 2018

[7] [7]

R. Khan, C. Krishnan and S. Sharma, Circuit Complexity in Fermionic Field Theory , Phys. Rev. D 98 (2018) 126001 [ 1801.07620]

work page arXiv 2018

[8] [8]

Circuit complexity for free fermions

L. Hackl and R.C. Myers, Circuit complexity for free fermions , JHEP 07 (2018) 139 [1803.10638]

work page internal anchor Pith review Pith/arXiv arXiv 2018

[9] [9]

Belin, R.C

A. Belin, R.C. Myers, S.-M. Ruan, G. S´ arosi and A.J. Speranza, Does Complexity Equal Anything?, Phys. Rev. Lett. 128 (2022) 081602 [ 2111.02429]

work page arXiv 2022

[10] [10]

Belin, R.C

A. Belin, R.C. Myers, S.-M. Ruan, G. S´ arosi and A.J. Speranza, Complexity equals anything II , JHEP 01 (2023) 154 [ 2210.09647]

work page arXiv 2023

[11] [11]

Yang and A.R

J. Yang and A.R. Frey, Complexity, scaling, and a phase transition , JHEP 09 (2023) 029 [2307.08229]

work page arXiv 2023

[12] [12]

Addendum to Computational Complexity and Black Hole Horizons

L. Susskind, Computational Complexity and Black Hole Horizons , Fortsch. Phys. 64 (2016) 24 [1403.5695]

work page internal anchor Pith review Pith/arXiv arXiv 2016

[13] [13]

Complexity and Shock Wave Geometries

D. Stanford and L. Susskind, Complexity and Shock Wave Geometries , Phys. Rev. D 90 (2014) 126007 [1406.2678]

work page internal anchor Pith review Pith/arXiv arXiv 2014

[14] [14]

Noether charge, black hole volume, and complexity

J. Couch, W. Fischler and P.H. Nguyen, Noether charge, black hole volume, and complexity , JHEP 03 (2017) 119 [ 1610.02038]

work page internal anchor Pith review Pith/arXiv arXiv 2017

[15] [15]

Complexity Equals Action

A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Holographic Complexity Equals Bulk Action? , Phys. Rev. Lett. 116 (2016) 191301 [ 1509.07876]

work page internal anchor Pith review Pith/arXiv arXiv 2016

[16] [16]

Complexity, action, and black holes

A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Complexity, action, and black holes, Phys. Rev. D 93 (2016) 086006 [ 1512.04993]

work page internal anchor Pith review Pith/arXiv arXiv 2016

[17] [17]

Engelhardt and r

N. Engelhardt and r. Folkestad, General bounds on holographic complexity , JHEP 01 (2022) 040 [2109.06883]

work page arXiv 2022

[18] [18]

Engelhardt and r

N. Engelhardt and r. Folkestad, Negative complexity of formation: the compact dimensions strike back, JHEP 07 (2022) 031 [ 2111.14897]

work page arXiv 2022

[19] [19]

Frey, Holographic complexity in string and M theory , Phys

A.R. Frey, Holographic complexity in string and M theory , Phys. Rev. D 111 (2025) 046011 [2410.21362]

work page arXiv 2025

[20] [20]

Supergravity and a Confining Gauge Theory: Duality Cascades and $\chi$SB-Resolution of Naked Singularities

I.R. Klebanov and M.J. Strassler, Supergravity and a confining gauge theory: Duality cascades and chi SB resolution of naked singularities , JHEP 08 (2000) 052 [ hep-th/0007191]. – 23 –

work page internal anchor Pith review Pith/arXiv arXiv 2000

[21] [21]

The AdS/CFT Correspondence and a New Positive Energy Conjecture for General Relativity

G.T. Horowitz and R.C. Myers, The AdS / CFT correspondence and a new positive energy conjecture for general relativity, Phys. Rev. D 59 (1998) 026005 [ hep-th/9808079]

work page internal anchor Pith review Pith/arXiv arXiv 1998

[22] [22]

Anti-de Sitter Space, Thermal Phase Transition, And Confinement In Gauge Theories

E. Witten, Anti-de Sitter space, thermal phase transition, and confinement in gauge theories , Adv. Theor. Math. Phys. 2 (1998) 505 [ hep-th/9803131]

work page internal anchor Pith review Pith/arXiv arXiv 1998

[23] [23]

Complexity of the AdS Soliton

A.P. Reynolds and S.F. Ross, Complexity of the AdS Soliton , Class. Quant. Grav. 35 (2018) 095006 [1712.03732]

work page internal anchor Pith review Pith/arXiv arXiv 2018

[24] [24]

The black hole in the throat - thermodynamics of strongly coupled cascading gauge theories

O. Aharony, A. Buchel and P. Kerner, The Black hole in the throat: Thermodynamics of strongly coupled cascading gauge theories, Phys. Rev. D 76 (2007) 086005 [ 0706.1768]

work page internal anchor Pith review Pith/arXiv arXiv 2007

[25] [25]

Buchel, Klebanov-Strassler black hole, JHEP 01 (2019) 207 [ 1809.08484]

A. Buchel, Klebanov-Strassler black hole, JHEP 01 (2019) 207 [ 1809.08484]

work page arXiv 2019

[26] [26]

Buchel, A bestiary of black holes on the conifold with fluxes , JHEP 06 (2021) 102 [2103.15188]

A. Buchel, A bestiary of black holes on the conifold with fluxes , JHEP 06 (2021) 102 [2103.15188]

work page arXiv 2021

[27] [27]

Complexity of Formation in Holography

S. Chapman, H. Marrochio and R.C. Myers, Complexity of Formation in Holography , JHEP 01 (2017) 062 [ 1610.08063]

work page internal anchor Pith review Pith/arXiv arXiv 2017

[28] [28]

Symmetry Breaking and Axionic Strings in the Warped Deformed Conifold

S.S. Gubser, C.P. Herzog and I.R. Klebanov, Symmetry breaking and axionic strings in the warped deformed conifold, JHEP 09 (2004) 036 [ hep-th/0405282]

work page internal anchor Pith review Pith/arXiv arXiv 2004

[29] [29]

The Baryonic Branch of Klebanov-Strassler Solution: a Supersymmetric Family of SU(3) Structure Backgrounds

A. Butti, M. Grana, R. Minasian, M. Petrini and A. Zaffaroni, The Baryonic branch of Klebanov-Strassler solution: A supersymmetric family of SU(3) structure backgrounds , JHEP 03 (2005) 069 [ hep-th/0412187]

work page internal anchor Pith review Pith/arXiv arXiv 2005

[30] [30]

Remarks on the Warped Deformed Conifold

C.P. Herzog, I.R. Klebanov and P. Ouyang, Remarks on the warped deformed conifold , in Modern Trends in String Theory: 2nd Lisbon School on g Theory Superstrings , 8, 2001 [hep-th/0108101]

work page internal anchor Pith review Pith/arXiv arXiv 2001

[31] [31]

Gravity Duals of Supersymmetric SU(N) x SU(N+M) Gauge Theories

I.R. Klebanov and A.A. Tseytlin, Gravity duals of supersymmetric SU(N) x SU(N+M) gauge theories, Nucl. Phys. B 578 (2000) 123 [ hep-th/0002159]

work page internal anchor Pith review Pith/arXiv arXiv 2000

[32] [32]

Complex geometry of conifolds and 5-brane wrapped on 2-sphere

G. Papadopoulos and A.A. Tseytlin, Complex geometry of conifolds and five-brane wrapped on two sphere, Class. Quant. Grav. 18 (2001) 1333 [ hep-th/0012034]

work page internal anchor Pith review Pith/arXiv arXiv 2001

[33] [33]

Tricomi, Asymptotische Eigenschaften der unvollst¨ andigen Gammafunktion, Math

F.G. Tricomi, Asymptotische Eigenschaften der unvollst¨ andigen Gammafunktion, Math. Z. 53 (1950) 136

work page 1950

[34] [34]

Tricomi, Sulla funzione gamma incompleta , Ann

F.G. Tricomi, Sulla funzione gamma incompleta , Ann. Mat. Pura Appl. 31 (1950) 263–279

work page 1950

[35] [35]

B¨ ohmer, Paul Eugen,Differenzengleichungen und bestimmte Integrale, Koehler, Liepzig (1939)

work page 1939

[36] [36]

Gravitational action with null boundaries

L. Lehner, R.C. Myers, E. Poisson and R.D. Sorkin, Gravitational action with null boundaries , Phys. Rev. D 94 (2016) 084046 [ 1609.00207]

work page internal anchor Pith review Pith/arXiv arXiv 2016

[37] [37]

Agarwal and A.R

N. Agarwal and A.R. Frey, work in progress,

work page

[38] [38]

Fatemiabhari and C

A. Fatemiabhari and C. Nunez, From conformal to confining field theories using holography , JHEP 03 (2024) 160 [ 2401.04158]

work page arXiv 2024

[39] [39]

Jørstad, R.C

E. Jørstad, R.C. Myers and S.-M. Ruan, Holographic complexity in dS d+1, JHEP 05 (2022) 119 [2202.10684]

work page arXiv 2022

[40] [40]

de Sitter Vacua in String Theory

S. Kachru, R. Kallosh, A.D. Linde and S.P. Trivedi, De Sitter vacua in string theory , Phys. Rev. D 68 (2003) 046005 [ hep-th/0301240]. – 24 –

work page internal anchor Pith review Pith/arXiv arXiv 2003