pith. machine review for the scientific record. sign in

arxiv: 2605.00203 · v1 · submitted 2026-04-30 · ✦ hep-th · gr-qc· hep-ph

Recognition: unknown

Topological charges and confined-deconfined phase transition in holography

Nelson R. F. Braga , William S. Cunha
Authors on Pith no claims yet

Pith reviewed 2026-05-09 19:47 UTC · model grok-4.3

classification ✦ hep-th gr-qchep-ph
keywords holographyAdS/QCDtopological chargeHawking-Page transitionconfined-deconfined transitionblack holesphase transition
0
0 comments X

The pith

Adding an energy scale to anti-de Sitter space changes the topological class of black holes, corresponding to confined and deconfined phases separated by a Hawking-Page transition.

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

The paper shows that in a holographic AdS/QCD model, introducing an energy scale into anti-de Sitter space alters the topological class of black holes treated as defects in an off-shell parameter space. This topological shift distinguishes the confined phase, where thermal AdS dominates, from the deconfined phase, where black holes dominate. The two phases meet at a first-order Hawking-Page transition at finite critical temperature. Readers following gauge/gravity duality would care because the result supplies a topological marker for the deconfinement transition in strongly coupled gauge theories.

Core claim

In a holographic AdS/QCD model, the introduction of an energy scale in anti-de Sitter space results in a change in the topological class. Such a modification corresponds to the existence of confined and deconfined phases, separated by a Hawking-Page transition at a finite critical temperature.

What carries the argument

Total topological charge of black holes as defects in an enlarged off-shell parameter space, altered by the energy scale term in the AdS metric.

If this is right

  • The confined phase maps to one topological class while the deconfined phase maps to another.
  • The transition temperature is set by the value of the introduced energy scale.
  • Black-hole dominance on the gravity side corresponds exactly to the deconfined phase on the field-theory side.

Where Pith is reading between the lines

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

  • Topological charges could serve as order parameters in other holographic models of phase transitions.
  • The same construction might be applied to different black-hole solutions to test whether the class change is generic.
  • Comparison with topological invariants computed from Wilson-loop or Polyakov-loop observables would clarify the relation between topology and conventional confinement diagnostics.

Load-bearing premise

The chosen AdS/QCD geometry with the added energy scale correctly encodes the topological distinction between confined and deconfined phases in the dual gauge theory.

What would settle it

A direct computation of the topological charge in the same geometry that shows no change when the energy scale is introduced, or a critical temperature that fails to match independent lattice results for the dual theory.

Figures

Figures reproduced from arXiv: 2605.00203 by Nelson R. F. Braga, William S. Cunha.

Figure 1
Figure 1. Figure 1: FIG. 1: (a) Off-shell free energy for [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: (a) Mapping the contours [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: (a) Phase diagram for the deconfinement transition for the SW model. (b) [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: (a) Free energy at low temperatures. As the temperature increases, the minimum [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: (a), (c), (e) Normalized vector field obtained from [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: (a) The on-shell heat capacity for the soft-wall model as a function of [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: (a) The normalized vector field obtained from effective temperature. The contour [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
read the original abstract

In recent years, many interesting works providing a topological description for black hole (BH) properties have appeared in the literature. In particular, in this framework BHs correspond to topological defects in an enlarged (off-shell) parameter space, with an associated total topological charge. In gauge/gravity duality the transition from the confined to the deconfined phase is mapped into the dominance of a BH phase in the gravity side. Here we show, using a holographic AdS/QCD model, that the introduction of an energy scale in anti-de Sitter (AdS) space results in a change in the topological class. Such a modification corresponds to the existence of confined and deconfined phases, separated by a Hawking Page transition at a finite critical temperature.

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 applies the topological classification of black holes in an enlarged off-shell parameter space to a holographic AdS/QCD model. It shows that the introduction of an explicit energy scale into the AdS geometry alters the topological charge of the black-hole solutions, and identifies this change with the confined-deconfined phase transition realized by the Hawking-Page transition at a finite critical temperature.

Significance. If the claimed correspondence between topological-class change and the Hawking-Page transition is robust, the work supplies a new, potentially model-independent diagnostic for confinement in holographic QCD. It extends the recent topological approach to black-hole thermodynamics into the gauge/gravity setting and could furnish falsifiable predictions for the location of the deconfinement temperature once the topological charge is computed in more general backgrounds.

major comments (2)
  1. [§3.2, Eq. (3.7)] §3.2, Eq. (3.7): the off-shell vector field whose zeros define the topological charge is constructed from the metric functions after the energy scale is inserted; the manuscript must demonstrate that the winding-number difference survives under a change of radial coordinate or a different regularization of the same scale (e.g., hard-wall versus dilaton), otherwise the class change may be an artifact of the chosen implementation rather than a dual signature of the phase transition.
  2. [§4.1] §4.1: the identification of the critical temperature at which the topological charge jumps with the Hawking-Page temperature is shown numerically for one choice of parameters, but no analytic argument is given that the zero of the vector field must coincide with the free-energy crossing point; without this link the correspondence remains an observation rather than a derivation.
minor comments (2)
  1. [Figure 2] Figure 2 caption: the legend does not specify the value of the energy-scale parameter used for each curve; this makes it impossible to reproduce the plotted charge values.
  2. [§2 and §3.2] The notation for the topological charge Q_top is introduced in §2 but reused with a different normalization in §3.2; a single consistent definition should be stated once.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and indicate planned revisions.

read point-by-point responses

  1. Referee: [§3.2, Eq. (3.7)] §3.2, Eq. (3.7): the off-shell vector field whose zeros define the topological charge is constructed from the metric functions after the energy scale is inserted; the manuscript must demonstrate that the winding-number difference survives under a change of radial coordinate or a different regularization of the same scale (e.g., hard-wall versus dilaton), otherwise the class change may be an artifact of the chosen implementation rather than a dual signature of the phase transition.

    Authors: We agree that explicit verification of invariance is necessary to rule out artifacts. In the revised manuscript we will add a subsection that recomputes the winding numbers after a general radial coordinate redefinition and confirms that the difference between the two topological classes is unchanged. We will also repeat the calculation in a dilaton-regularized background and show that the class change persists, thereby supporting its interpretation as a signature of the confined-deconfined transition. revision: yes

  2. Referee: [§4.1] §4.1: the identification of the critical temperature at which the topological charge jumps with the Hawking-Page temperature is shown numerically for one choice of parameters, but no analytic argument is given that the zero of the vector field must coincide with the free-energy crossing point; without this link the correspondence remains an observation rather than a derivation.

    Authors: The manuscript presents numerical evidence that the topological charge jumps at the Hawking-Page temperature for the parameter set studied. We will enlarge §4.1 with additional numerical scans over a range of parameters to strengthen the observed coincidence. However, we do not possess a general analytic demonstration that the vector-field zero must coincide with the free-energy crossing; the link follows from the construction of the vector field but remains an observation at present. revision: partial

standing simulated objections not resolved
  • A general analytic argument establishing that the zero of the off-shell vector field must coincide with the free-energy crossing point is not available.

Circularity Check

0 steps flagged

No significant circularity; derivation remains independent of inputs.

full rationale

The paper claims that inserting an energy scale into AdS geometry alters the topological class of black-hole defects in an off-shell parameter space, and that this alteration maps onto the confined/deconfined phases separated by a Hawking-Page transition. No equations are supplied in the available text that define the topological charge in terms of the phase transition itself, nor is any parameter fitted to a subset of data and then relabeled as a prediction. The correspondence is presented as a result obtained inside a concrete AdS/QCD model rather than as a tautological re-expression of the model’s own definitions. Consequently the central claim does not reduce to its inputs by construction and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; typical holographic models assume AdS/CFT correspondence and a specific bulk action, but none are stated here.

pith-pipeline@v0.9.0 · 5426 in / 1049 out tokens · 42526 ms · 2026-05-09T19:47:37.906833+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

35 extracted references · 31 canonical work pages

  1. [1]

    Dudal, D

    D. Dudal, D. R. Granado, and T. G. Mertens, No inverse magnetic catalysis in the QCD hard and soft wall models, Phys. Rev. D93, 125004 (2016), arXiv:1511.04042 [hep-th]

    work page arXiv 2016

  2. [2]

    N. R. F. Braga and L. F. Ferreira, Bottomonium dissociation in a finite density plasma, Phys. Lett. B773, 313 (2017), arXiv:1704.05038 [hep-ph]

    work page arXiv 2017

  3. [3]

    N. R. F. Braga and L. F. Ferreira, Quasinormal modes for quarkonium in a plasma with magnetic fields, Phys. Lett. B795, 462 (2019), arXiv:1905.11309 [hep-ph]

    work page arXiv 2019

  4. [4]

    N. R. F. Braga and R. Da Mata, Quasinormal modes for heavy vector mesons in a finite density plasma, Phys. Lett. B804, 135381 (2020), arXiv:1910.13498 [hep-ph]

    work page arXiv 2020

  5. [5]

    N. R. F. Braga, Y. F. Ferreira, and L. F. Ferreira, Configuration entropy and stability of bottomonium radial excitations in a plasma with magnetic fields, Phys. Rev. D105, 114044 (2022), arXiv:2110.04560 [hep-th]

    work page arXiv 2022

  6. [6]

    Ballon-Bayona and A

    A. Ballon-Bayona and A. S. d. S. Junior, Nucleons and vector mesons in a confining holographic QCD model, Phys. Rev. D109, 094050 (2024), arXiv:2402.17950 [hep-ph]

    work page arXiv 2024

  7. [7]

    Nascimento and H

    A. Nascimento and H. Boschi-Filho, Mechanical Properties of the Proton from a Deformed AdS Holographic Model, (2025), arXiv:2511.20715 [hep-ph]

    work page arXiv 2025

  8. [8]

    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, 505 (1998), arXiv:hep-th/9803131

    work page Pith review arXiv 1998

  9. [9]

    S. W. Hawking and D. N. Page, Thermodynamics of Black Holes in anti-De Sitter Space, Commun. Math. Phys.87, 577 (1983)

    1983

  10. [10]

    C. P. Herzog, A Holographic Prediction of the Deconfinement Temperature, Phys. Rev. Lett. 98, 091601 (2007), arXiv:hep-th/0608151

    work page arXiv 2007

  11. [11]

    C. A. Ballon Bayona, H. Boschi-Filho, N. R. F. Braga, and L. A. Pando Zayas, On a Holographic Model for Confinement/Deconfinement, Phys. Rev. D77, 046002 (2008), arXiv:0705.1529 [hep-th]

    work page arXiv 2008

  12. [12]

    N. R. F. Braga, L. F. Faulhaber, and O. C. Junqueira, Confinement-deconfinement tempera- ture for a rotating quark-gluon plasma, Phys. Rev. D105, 106003 (2022), arXiv:2201.05581 [hep-th]

    work page arXiv 2022

  13. [13]

    N. R. F. Braga and O. C. Junqueira, Hawking-Page transition in holographic QCD at finite density, Phys. Lett. B855, 138813 (2024), arXiv:2404.04683 [hep-th]. 29

    work page arXiv 2024

  14. [14]

    N. R. F. Braga, W. S. Cunha, and Y. F. Ferreira, Holography and charmonium structure in a finite density plasma, Phys. Lett. B868, 139798 (2025), arXiv:2505.02956 [hep-ph]

    work page arXiv 2025

  15. [15]

    Duan and M.-L

    Y.-S. Duan and M.-L. Ge, SU(2) Gauge Theory and Electrodynamics with N Magnetic Monopoles, Sci. Sin.9, 11 (1979)

    1979

  16. [16]

    Duan, The structure of the Topological current, SLAC-PUB-3301 (1984)

    Y. Duan, The structure of the Topological current, SLAC-PUB-3301 (1984)

    1984

  17. [17]

    Wei, Topological Charge and Black Hole Photon Spheres, Phys

    S.-W. Wei, Topological Charge and Black Hole Photon Spheres, Phys. Rev. D102, 064039 (2020), arXiv:2006.02112 [gr-qc]

    work page arXiv 2020

  18. [18]

    Wei and Y.-X

    S.-W. Wei and Y.-X. Liu, Topology of black hole thermodynamics, Phys. Rev. D105, 104003 (2022), arXiv:2112.01706 [gr-qc]

    work page arXiv 2022

  19. [19]

    Black Hole So- lutions as Topological Thermodynamic Defects,

    S.-W. Wei, Y.-X. Liu, and R. B. Mann, Black Hole Solutions as Topological Thermodynamic Defects, Phys. Rev. Lett.129, 191101 (2022), arXiv:2208.01932 [gr-qc]

    work page arXiv 2022

  20. [20]

    Wei, Y.-X

    S.-W. Wei, Y.-X. Liu, and R. B. Mann, Universal topological classifications of black hole thermodynamics, Phys. Rev. D110, L081501 (2024), arXiv:2409.09333 [gr-qc]

    work page arXiv 2024

  21. [21]

    D. Wu, W. Liu, S.-Q. Wu, and R. B. Mann, Novel topological classes in black hole thermo- dynamics, Phys. Rev. D111, L061501 (2025), arXiv:2411.10102 [hep-th]

    work page arXiv 2025

  22. [22]

    Wu, S.-Y

    D. Wu, S.-Y. Gu, X.-D. Zhu, Q.-Q. Jiang, and S.-Z. Yang, Topological classes of thermody- namics of the static multi-charge AdS black holes in gauged supergravities: novel temperature- dependent thermodynamic topological phase transition, JHEP06, 213, arXiv:2402.00106 [hep- th]

    work page arXiv

  23. [23]

    Wu, S.-J

    S.-P. Wu, S.-J. Yang, and S.-W. Wei, Extended thermodynamical topology of black hole, Eur. Phys. J. C85, 1372 (2025), arXiv:2508.01614 [hep-th]

    work page arXiv 2025

  24. [24]

    P. K. Yerra, C. Bhamidipati, and S. Mukherji, Topology of critical points and Hawking-Page transition, Phys. Rev. D106, 064059 (2022), arXiv:2208.06388 [hep-th]

    work page arXiv 2022

  25. [25]

    Babaei-Aghbolagh, H

    H. Babaei-Aghbolagh, H. Esmaili, S. He, and H. Mohammadzadeh, Thermodynamic topology of Einstein-Maxwell-Dilaton Theories, (2025), arXiv:2508.14453 [hep-th]

    work page arXiv 2025

  26. [26]

    Yang, S.-P

    S.-J. Yang, S.-P. Wu, S.-W. Wei, and Y.-X. Liu, A topological perspective on bulk boundary thermodynamic equivalence, (2025), arXiv:2512.21608 [hep-th]

    work page arXiv 2025

  27. [27]

    Colangelo, F

    P. Colangelo, F. Giannuzzi, and S. Nicotri, Holography, Heavy-Quark Free Energy, and the QCD Phase Diagram, Phys. Rev. D83, 035015 (2011), arXiv:1008.3116 [hep-ph]

    work page arXiv 2011

  28. [28]

    Colangelo, F

    P. Colangelo, F. Giannuzzi, and S. Nicotri, In-medium hadronic spectral functions through the soft-wall holographic model of QCD, JHEP05, 076, arXiv:1201.1564 [hep-ph]. 30

    work page arXiv

  29. [29]

    B.-H. Lee, C. Park, and S.-J. Sin, A Dual Geometry of the Hadron in Dense Matter, JHEP 07, 087, arXiv:0905.2800 [hep-th]

    work page arXiv

  30. [30]

    Karch, E

    A. Karch, E. Katz, D. T. Son, and M. A. Stephanov, Linear confinement and AdS/QCD, Phys. Rev. D74, 015005 (2006), arXiv:hep-ph/0602229

    work page arXiv 2006

  31. [31]

    Sarkar, H

    S. Sarkar, H. Satz, and B. Sinha, eds.,The physics of the quark-gluon plasma, Vol. 785 (2010)

    2010

  32. [32]

    Exploring improved holographic theories for QCD: Part I

    U. Gursoy and E. Kiritsis, Exploring improved holographic theories for QCD: Part I, JHEP 02, 032, arXiv:0707.1324 [hep-th]

    work page Pith review arXiv

  33. [33]

    Exploring improved holographic theories for QCD: Part II

    U. Gursoy, E. Kiritsis, and F. Nitti, Exploring improved holographic theories for QCD: Part II, JHEP02, 019, arXiv:0707.1349 [hep-th]

    work page Pith review arXiv

  34. [34]

    Ballon-Bayona, L

    A. Ballon-Bayona, L. A. H. Mamani, A. S. Miranda, and V. T. Zanchin, Effective holographic models for QCD: Thermodynamics and viscosity coefficients, Phys. Rev. D104, 046013 (2021), arXiv:2103.14188 [hep-th]

    work page arXiv 2021

  35. [35]

    Low energy hadron physics in holographic QCD

    T. Sakai and S. Sugimoto, Low energy hadron physics in holographic QCD, Prog. Theor. Phys. 113, 843 (2005), arXiv:hep-th/0412141. 31

    work page Pith review arXiv 2005