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arxiv: 2605.19687 · v1 · pith:I6GM3B2Jnew · submitted 2026-05-19 · 🌀 gr-qc

Compact objects in AdS spacetime with exponential, quadratic and power-law bosonic mass profiles

Samprity Das , Aroonkumar Beesham , Surajit Chattopadhyay This is my paper

Pith reviewed 2026-05-20 04:40 UTC · model grok-4.3

classification 🌀 gr-qc
keywords compact starsAdS spacetimeBose-Einstein condensatebosonic mass profilesenergy conditionsBuchdahl limitstellar stability
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The pith

Bosonic mass profiles in exponential, quadratic and power-law forms produce stable compact stellar configurations in AdS spacetime.

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

The paper studies compact objects formed from a zero-temperature Bose-Einstein condensate in anti-de Sitter spacetime. It assigns the bosonic mass three different radial dependences—exponential, quadratic, and power-law—and solves for the resulting mass, compactness, surface redshift, density, pressure, adiabatic index, and energy conditions. The computed mass grows steadily outward and stays inside observed compact-star ranges while the compactness ratio remains below Buchdahl’s bound, indicating equilibrium stellar models rather than collapse. Both the null and strong energy conditions hold everywhere inside the star, which the authors take as evidence of dynamical stability. The configurations also show greater mass concentration near the surface.

Core claim

When the bosonic mass inside an AdS compact object is given an exponential, quadratic, or power-law dependence on radius, the integrated mass increases monotonically with radius, remains within observational compact-star mass limits, and yields a compactness ratio below Buchdahl’s limit; the null and strong energy conditions are satisfied throughout the interior, confirming that the models describe stable compact stellar configurations rather than collapsing ones.

What carries the argument

The bosonic mass expressed as a chosen function of radial coordinate (exponential, quadratic, or power-law) that enters the metric and matter sector of the AdS Einstein equations for the condensate.

If this is right

  • All three profiles generate mass-radius curves that lie inside the range of observed compact-star masses.
  • Compactness for every profile stays safely below the Buchdahl upper bound.
  • Satisfaction of the null and strong energy conditions throughout the interior supports dynamical stability.
  • Mass density is higher near the surface than in the core for the adopted profiles.

Where Pith is reading between the lines

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

  • The same mass-function approach could be applied to other asymptotically AdS geometries to test how the choice of profile affects the approach to a black-hole limit.
  • Extending the static models to include slow rotation would allow direct comparison with pulsar timing or gravitational-wave data on compact objects.
  • Because the profiles are phenomenological, matching them to a holographic dual description of the condensate could provide a microscopic origin for the radial dependence.

Load-bearing premise

The bosonic mass is taken to follow one of three simple phenomenological functions of radius without a first-principles derivation from a microscopic theory.

What would settle it

A radial integration of the Einstein equations for any of the three mass profiles that produces a compactness exceeding Buchdahl’s limit or a region where the strong energy condition fails would falsify the stability conclusion.

Figures

Figures reproduced from arXiv: 2605.19687 by Aroonkumar Beesham, Samprity Das, Surajit Chattopadhyay.

Figure 1
Figure 1. Figure 1: FIG. 1: (a) Evolution of mass about radius [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: (a) Evolution of density about radius [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: (a) Evolution of Null Energy Condition about radius [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: (a) Evolution of Adiabatic Index about radius [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: (a) Evolution of mass with radius [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: (a) Evolution of density about radius [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: (a) Evolution of Null Energy Condition about radius [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: (a) Evolution of Adiabatic Index about radius [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: (a) Evolution of mass about radius [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Evolution of density about radius [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Evolution of density about radius [PITH_FULL_IMAGE:figures/full_fig_p027_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Evolution of the pressure about radius [PITH_FULL_IMAGE:figures/full_fig_p027_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Evolution of the pressure about radius [PITH_FULL_IMAGE:figures/full_fig_p028_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Evolution of the null energy conditions about radius [PITH_FULL_IMAGE:figures/full_fig_p028_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: Evolution of the null energy conditions about radius [PITH_FULL_IMAGE:figures/full_fig_p029_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: Evolution of the strong energy conditions about radius [PITH_FULL_IMAGE:figures/full_fig_p029_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: Evolution of the strong energy conditions about radius [PITH_FULL_IMAGE:figures/full_fig_p030_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18: Evolution of Adiabatic Index about radius [PITH_FULL_IMAGE:figures/full_fig_p030_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19: Evolution of Adiabatic Index about radius [PITH_FULL_IMAGE:figures/full_fig_p031_19.png] view at source ↗
read the original abstract

This paper reports a study on the formation and physical characteristics of compacts stars in AdS spacetime within the framework of Bose-Einstein Condensate. Considering a Bose-Einstein condensate background at zero temperature this study works on total mass, compactness, surface redshift, density, pressure, adiabatic index and energy conditions. The bosonic mass has been taken as three distinct functions of radial coordinate in exponential form, quadratic form, and power law form. Our results reveal that the mass increases monotonically with radius and remains within observational limit for all the observationally motivated compact-star mass scales considered in this study and the compactness for all the cases is within Buchdahl's limit and hence it was confirmed that the configuration correspondence to compact stellar configuration models rather than forming a collapsing model. Both NEC and SEC are satisfied throughout the stellar interior and hence dynamical stability is ensured. Furthermore, the study also confirms the enhanced mass concentration near the outer region in the stellar models under consideration. Hence present study explores the physical properties and stability of compact bosonic configurations in AdS spacetime within a holographically motivated framework. The present analysis is primarily phenomenological and qualitative in nature. The models considered here are intended to explore possible behaviours of self-gravitating bosonic configurations in AdS geometry and are not proposed as fully realistic neutron-star models.

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 / 3 minor

Summary. The paper examines compact stellar configurations in Anti-de Sitter (AdS) spacetime within a zero-temperature Bose-Einstein Condensate (BEC) framework. It adopts three phenomenological bosonic mass profiles m(r)—exponential, quadratic, and power-law—as functions of the radial coordinate, then computes total mass, compactness, surface redshift, density, pressure, adiabatic index, and energy conditions. The central claims are that mass increases monotonically and stays within observational limits, compactness respects Buchdahl's bound (confirming compact-star rather than collapsing models), and both NEC and SEC hold throughout the interior (ensuring dynamical stability), with enhanced mass concentration near the outer regions. The analysis is explicitly described as phenomenological and qualitative, not proposed as realistic neutron-star models.

Significance. If the mass profiles prove consistent with the Einstein equations sourced by a BEC fluid in AdS, the results provide qualitative exploration of possible self-gravitating bosonic configurations in a holographically motivated geometry, including confirmation of stability indicators and outer mass concentration. The work's value is limited by its phenomenological construction, offering illustrative behaviors rather than predictive or first-principles models.

major comments (2)
  1. [Mass profiles section] § on mass profiles (following abstract description of exponential, quadratic, and power-law forms): the three m(r) profiles are inserted directly as ad-hoc functional forms with free scale and exponent parameters, without derivation from the BEC equation of state (typically p = K ρ² for non-relativistic condensate) or explicit verification that the implied ρ(r), p(r) satisfy the modified TOV equation including the negative cosmological constant. This is load-bearing for the stability and energy-condition claims, as an inconsistent p-ρ relation would mean the configurations are not valid BEC solutions in AdS.
  2. [Energy conditions and stability results] Results section on energy conditions and stability: while NEC and SEC are stated to be satisfied, the manuscript must demonstrate how these are obtained from the assumed m(r) via the Einstein equations (including AdS term) and confirm that the profiles produce a self-consistent fluid source; without this, the conclusion that 'dynamical stability is ensured' rests on unverified assumptions.
minor comments (3)
  1. [Abstract] Abstract: 'the configuration correspondence to compact stellar configuration models' contains a grammatical error and should read 'the configurations correspond to compact stellar models'.
  2. [Parameter choices] Throughout: specify the numerical values or ranges chosen for the free parameters in each mass profile and discuss any sensitivity of the monotonic mass growth or Buchdahl compliance to these choices.
  3. [Notation] Notation: ensure consistent use of symbols for bosonic mass m(r), energy density, and pressure when transitioning between the three profile cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments on the consistency of the phenomenological mass profiles with the Einstein equations. We clarify that the study is explicitly phenomenological and qualitative, as stated in the abstract, and does not claim to derive the profiles from a specific BEC equation of state. We will revise the manuscript to include explicit derivations showing how the energy density and pressure are obtained from the assumed m(r) via the Einstein equations in AdS spacetime, thereby confirming self-consistency of the fluid sources and the validity of the energy-condition and stability results within this framework.

read point-by-point responses

  1. Referee: [Mass profiles section] § on mass profiles (following abstract description of exponential, quadratic, and power-law forms): the three m(r) profiles are inserted directly as ad-hoc functional forms with free scale and exponent parameters, without derivation from the BEC equation of state (typically p = K ρ² for non-relativistic condensate) or explicit verification that the implied ρ(r), p(r) satisfy the modified TOV equation including the negative cosmological constant. This is load-bearing for the stability and energy-condition claims, as an inconsistent p-ρ relation would mean the configurations are not valid BEC solutions in AdS.

    Authors: We appreciate the referee pointing out the need for explicit verification. Our analysis is phenomenological by design, as noted in the manuscript, and the chosen m(r) forms are intended to illustrate possible behaviors rather than to represent exact solutions derived from the BEC equation of state. In the revised manuscript, we will add a dedicated subsection deriving the energy density ρ(r) directly from the Einstein equations using the mass function m(r) and the AdS cosmological constant term. We will then obtain the pressure p(r) from the modified Tolman-Oppenheimer-Volkoff equation and confirm that the resulting p-ρ relation is consistent for each profile. This will explicitly demonstrate that the assumed forms produce valid fluid sources within the phenomenological setup. revision: yes

  2. Referee: [Energy conditions and stability results] Results section on energy conditions and stability: while NEC and SEC are stated to be satisfied, the manuscript must demonstrate how these are obtained from the assumed m(r) via the Einstein equations (including AdS term) and confirm that the profiles produce a self-consistent fluid source; without this, the conclusion that 'dynamical stability is ensured' rests on unverified assumptions.

    Authors: We agree that the derivations should be shown explicitly to support the claims. In the revised version, we will present the full expressions for the energy density and isotropic pressure obtained from the Einstein field equations with the negative cosmological constant, using each m(r) profile. We will then compute the null energy condition (ρ + p ≥ 0) and strong energy condition (ρ + 3p ≥ 0 and ρ + p ≥ 0) directly from these quantities and verify that they hold throughout the interior for the chosen parameter ranges. This will confirm that the profiles yield self-consistent sources and that the stability indicators follow from the field equations rather than from unverified assumptions. We will also emphasize that these results are valid within the stated phenomenological context. revision: yes

Circularity Check

0 steps flagged

No significant circularity; phenomenological inputs are explicit and non-reductive

full rationale

The paper states that the bosonic mass profiles are 'taken as' exponential, quadratic, and power-law functions of radius and frames the entire analysis as 'primarily phenomenological and qualitative in nature' with models 'intended to explore possible behaviours' rather than derived from the BEC equation of state. The reported outcomes (monotonic mass growth, compactness below Buchdahl's limit, satisfaction of NEC/SEC) are direct verifications of quantities obtained by inserting these assumed m(r) forms into the Einstein equations with negative cosmological constant; no step claims a first-principles derivation that reduces to the inputs by construction, no self-citation is load-bearing, and no parameter fit is relabeled as an independent prediction. The derivation chain is therefore self-contained against external benchmarks such as Buchdahl's limit and energy conditions.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard general-relativistic stellar structure plus the assumption of a zero-temperature BEC; the three mass profiles are additional modeling choices without independent microscopic justification.

free parameters (1)
  • scale and exponent parameters in the three mass profiles
    Exponential, quadratic, and power-law forms each introduce at least one free scale or power that must be chosen or adjusted to obtain acceptable stellar solutions.
axioms (2)
  • domain assumption Bose-Einstein condensate background at zero temperature
    Explicitly stated in the abstract as the starting point for the bosonic configurations.
  • domain assumption AdS spacetime geometry with holographic motivation
    The entire analysis is performed inside anti-de Sitter spacetime.

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Reference graph

Works this paper leans on

54 extracted references · 54 canonical work pages · 6 internal anchors

  1. [1]

    Higher adiabatic index values show that pressure rises quickly with density, making the star model more resilient to compression and producing a more stable structure

    The adiabatic index is crucial for understanding how changes in density affect the changes in pressure. Higher adiabatic index values show that pressure rises quickly with density, making the star model more resilient to compression and producing a more stable structure. The graph displays a monotonic increase in pattern within the interior of the stars, ...

    work page

  2. [2]

    top-down

    proposed this limiting value to evaluate the dynamical stability of anisotropic realistic stars with spherical symmetry in the presence of small perturbations in radial adiabatic. Heintzmann and Hillebrandt [45] state that for compact objects with a rising anisotropy factor, the relativistic stability requirement, Γ, should be greater than 4/3. Fig. 4 and...

    work page

  3. [3]

    S. L. Shapiro and S. A. Teukolsky, Black holes, white dwarfs and neutron stars: the physics of compact objects (John Wiley & Sons, 2024)

    work page 2024

  4. [4]

    Sedrakian, J

    A. Sedrakian, J. J. Li, and F. Weber, Progress in Particle and Nuclear Physics 131, 104041 (2023). 36

    work page 2023

  5. [5]

    S. L. Shapiro, S. A. Teukolsky, B. Holes, W. Dwarfs, and N. Stars, Wiley, New York 19832, 119 (1983)

    work page 1983

  6. [6]

    Weber, Pulsars as astrophysical laboratories for nuclear and particle physics (Routledge, 2017)

    F. Weber, Pulsars as astrophysical laboratories for nuclear and particle physics (Routledge, 2017)

    work page 2017

  7. [7]

    Einstein et al., Annalen Phys 49, 769 (1916)

    A. Einstein et al., Annalen Phys 49, 769 (1916)

    work page 1916

  8. [8]

    H. A. Lorentz, A. Einstein, H. Minkowski, H. Weyl, and A. Sommerfeld, The principle of relativity: a collection of original memoirs on the special and general theory of relativity (Courier Corporation, 1952)

    work page 1952

  9. [9]

    P. A. M. Dirac, General theory of relativity , vol. 14 (Princeton University Press, 1996)

    work page 1996

  10. [10]

    Maldacena, International journal of theoretical physics 38, 1113 (1999)

    J. Maldacena, International journal of theoretical physics 38, 1113 (1999)

    work page 1999

  11. [11]

    Burikham and T

    P. Burikham and T. Chullaphan, Journal of High Energy Physics 2014, 1 (2014)

    work page 2014

  12. [12]

    Dimensional Reduction in Quantum Gravity

    G. Hooft, arXiv preprint gr-qc/9310026 (1993)

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  13. [13]

    Anti De Sitter Space And Holography

    E. Witten, arXiv preprint hep-th/9802150 (1998)

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  14. [14]

    Kovensky, A

    N. Kovensky, A. Poole, and A. Schmitt, Physical Review D 105, 034022 (2022)

    work page 2022

  15. [15]

    Kovensky, A

    N. Kovensky, A. Poole, and A. Schmitt, SciPost Physics Proceedings p. 019 (2022)

    work page 2022

  16. [16]

    de Boer, K

    J. de Boer, K. Papadodimas, and E. Verlinde, Journal of High Energy Physics 2010, 1 (2010)

    work page 2010

  17. [17]

    Arsiwalla, J

    X. Arsiwalla, J. de Boer, K. Papadodimas, and E. Verlinde, Journal of High Energy Physics 2011, 1 (2011)

    work page 2011

  18. [18]

    A Gravity Dual of RHIC Collisions

    E. Shuryak, S.-J. Sin, and I. Zahed, arXiv preprint hep-th/0511199 (2005)

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  19. [19]

    P. M. Chesler and L. G. Yaffe, arXiv preprint arXiv:0812.2053 (2008)

    work page internal anchor Pith review Pith/arXiv arXiv 2053

  20. [20]

    Bhattacharyya and S

    S. Bhattacharyya and S. Minwalla, Journal of High Energy Physics 2009, 034 (2009)

    work page 2009

  21. [21]

    Balasubramanian, A

    V. Balasubramanian, A. Bernamonti, J. de Boer, N. Copland, B. Craps, E. Keski-Vakkuri, B. Müller, A. Schäfer, M. Shigemori, and W. Staessens, Physical review letters 106, 191601 (2011)

    work page 2011

  22. [22]

    S. Das, P. Rudra, and S. Chattopadhyay, Nucl. Phys. B 1012, 9 (2025)

    work page 2025

  23. [23]

    Ji and S.-J

    S. Ji and S.-J. Sin, Physical Review D 50, 3655 (1994)

    work page 1994

  24. [24]

    W. Hu, R. Barkana, and A. Gruzinov, Physical Review Letters 85, 1158 (2000)

    work page 2000

  25. [25]

    Observational constraints on holographic dark energy with varying gravitational constant

    Z. Li and C. Bambi, arXiv: 0912.0923 p. 031 (2010)

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  26. [26]

    V. H. Robles and T. Matos, Monthly Notices of the Royal Astronomical Society 422, 282 (2012)

    work page 2012

  27. [27]

    Chavanis and L

    P.-H. Chavanis and L. Delfini, Physical Review D—Particles, Fields, Gravitation, and Cosmology 84, 043532 (2011)

    work page 2011

  28. [28]

    Witten, Communications in mathematical physics 121, 351 (1989)

    E. Witten, Communications in mathematical physics 121, 351 (1989)

    work page 1989

  29. [29]

    Feng, arXiv preprint arXiv:2401.01617 (2024)

    T. Feng, arXiv preprint arXiv:2401.01617 (2024)

    work page arXiv 2024

  30. [30]

    Hoyos, N

    C. Hoyos, N. Jokela, and A. Vuorinen, Progress in Particle and Nuclear Physics 126, 103972 (2022)

    work page 2022

  31. [31]

    Hartmann, B

    B. Hartmann, B. Kleihaus, J. Kunz, and I. Schaffer, Physical Review D—Particles, Fields, Gravitation, and Cosmology 88, 124033 (2013)

    work page 2013

  32. [32]

    Wachter, Relativistic quantum mechanics , vol

    A. Wachter, Relativistic quantum mechanics , vol. 422 (Springer, 2011)

    work page 2011

  33. [33]

    Bhuiyan, in Journal of Physics: Conference Series (IOP Publishing, 2021), vol

    G. Bhuiyan, in Journal of Physics: Conference Series (IOP Publishing, 2021), vol. 1718, p. 012006

    work page 2021

  34. [34]

    P. Bhar, K. Singh, and N. Pant, Indian Journal of Physics 91, 701 (2017)

    work page 2017

  35. [35]

    Moraes, P

    P. Moraes, P. K. Sahoo, S. S. Kulkarni, and S. Agarwal, Chinese Physics Letters 36, 120401 (2019). 37

    work page 2019

  36. [36]

    S. Das, I. Radinschi, and S. Chattopadhyay, Axioms 12, 234 (2023)

    work page 2023

  37. [37]

    S. Das, A. Beesham, and S. Chattopadhyay, Annals of Physics 458, 169460 (2023)

    work page 2023

  38. [38]

    S. Das, S. Chattopadhyay, and E. Güdekli, Nuclear Physics B p. 117072 (2025)

    work page 2025

  39. [39]

    B. V. Ivanov, Physical Review D 65, 104011 (2002)

    work page 2002

  40. [40]

    Das and S

    S. Das and S. Chattopadhyay, Astroparticle Physics 165, 103053 (2025)

    work page 2025

  41. [41]

    Morales and F

    E. Morales and F. Tello-Ortiz, The European Physical Journal C 78, 618 (2018)

    work page 2018

  42. [42]

    S. Das, E. Güdekli, and S. Chattopadhyay, International Journal of Modern Physics A 41, 2650060 (2026)

    work page 2026

  43. [43]

    Hawking, Communications in Mathematical Physics 46, 206 (1976)

    S. Hawking, Communications in Mathematical Physics 46, 206 (1976)

    work page 1976

  44. [44]

    Santos, J

    J. Santos, J. Alcaniz, M. Reboucas, and F. Carvalho, Physical Review D—Particles, Fields, Gravitation, and Cosmology 76, 083513 (2007)

    work page 2007

  45. [45]

    Balart and E

    L. Balart and E. C. Vagenas, Physics Letters B 730, 14 (2014)

    work page 2014

  46. [46]

    Das and S

    S. Das and S. Chattopadhyay, Physica Scripta 99, 055020 (2024)

    work page 2024

  47. [47]

    Heintzmann and W

    H. Heintzmann and W. Hillebrandt, Astronomy and Astrophysics, vol. 38, no. 1, Jan. 1975, p. 51-55. Research supported by the Deutsche Forschungsgemeinschaft. 38, 51 (1975)

    work page 1975

  48. [48]

    Chandrasekhar, Physical Review Letters 12, 114 (1964)

    S. Chandrasekhar, Physical Review Letters 12, 114 (1964)

    work page 1964

  49. [49]

    R. S. (London), Proceedings of the Royal Society , vol. 17 (1869)

    work page

  50. [50]

    Nuclear pasta in supernovae and neutron stars

    G. Watanabe and T. Maruyama, arXiv preprint arXiv:1109.3511 (2011)

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  51. [51]

    Vogt and P

    D. Vogt and P. Letelier, Monthly Notices of the Royal Astronomical Society 402, 1313 (2010)

    work page 2010

  52. [52]

    S. S. Gubser, I. R. Klebanov, and A. Peet, Physical Review D 54, 3915 (1996)

    work page 1996

  53. [53]

    I. R. Klebanov, Nuclear Physics B 496, 231 (1997)

    work page 1997

  54. [54]

    S. S. Gubser and I. R. Klebanov, Physics Letters B 413, 41 (1997)

    work page 1997