Exactly solvable gravitating perfect fluid solitons in (2 + 1) dimensions

  • C. AdamEmail author
  • T. Romanczukiewicz
  • M. Wachla
  • A. Wereszczynski
Open Access
Regular Article - Theoretical Physics


The Bogomolnyi-Prasad-Sommerfield (BPS) baby Skyrme model coupled to gravity is considered. We show that in an asymptotically flat space-time the model still possesses the BPS property, i.e., admits a BPS reduction to first order Bogomolnyi equations, which guarantees that the corresponding proper energy is a linear function of the topological charge. We also find the mass-radius relation as well as the maximal mass and radius. All these results are obtained in an analytical manner, which implies the complete solvability of this selfgravitating matter system.

If a cosmological constant is added, then the BPS property is lost. In de Sitter (dS ) space-time both extremal and non-extremal solutions are found, where the former correspond to finite positive pressure solutions of the flat space-time model. For the asymptotic anti-de Sitter (AdS ) case, extremal solutions do not exist as there are no negative pressure BPS baby Skyrmions in flat space-time. Non-extremal solutions with AdS asymptotics do exist and may be constructed numerically. The impact of the negative cosmological constant on the mass-radius relation is studied. We also found two potentials for which exact multi-soliton solutions in the external AdS space can be obtained. Finally, we elaborate on the implications of these findings for certain three-dimensional models of holographic QCD.


Field Theories in Lower Dimensions Solitons Monopoles and Instantons Holography and quark-gluon plasmas Integrable Field Theories 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    T.H.R. Skyrme, A non-linear field theory, Proc. Roy. Soc. Lond. 260 (1961) 127.MathSciNetCrossRefzbMATHADSGoogle Scholar
  2. [2]
    T.H.R. Skyrme, A unified field theory of mesons and baryons, Nucl. Phys. 31 (1962) 556.MathSciNetCrossRefGoogle Scholar
  3. [3]
    B.M. A.G. Piette, B.J. Schroers and W.J. Zakrzewski, Multi-solitons in a two-dimensional Skyrme model, Z. Phys. C 65 (1995) 165 [hep-th/9406160] [INSPIRE].ADSGoogle Scholar
  4. [4]
    B.M. A.G. Piette, B.J. Schroers and W.J. Zakrzewski, Dynamics of baby skyrmions, Nucl. Phys. B 439 (1995) 205 [hep-ph/9410256] [INSPIRE].
  5. [5]
    R.S. Ward, Planar Skyrmions at high and low density, Nonlinearity 17 (2004) 1033.MathSciNetCrossRefzbMATHADSGoogle Scholar
  6. [6]
    I. Hen and M. Karliner, Rotational symmetry breaking in baby Skyrme models, Nonlinearity 21 (2008) 399 [arXiv:0710.3939].MathSciNetCrossRefzbMATHADSGoogle Scholar
  7. [7]
    D. Foster, Baby Skyrmion chains, Nonlinearity 23 (2010) 465.MathSciNetCrossRefzbMATHADSGoogle Scholar
  8. [8]
    J. Jaykka, M. Speight and P. Sutcliffe, Broken Baby Skyrmions, Proc. Roy. Soc. Lond. A 468 (2012) 1085 [arXiv:1106.1125] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  9. [9]
    M. Kobayashi and M. Nitta, Fractional vortex molecules and vortex polygons in a baby Skyrme model, Phys. Rev. D 87 (2013) 125013 [arXiv:1307.0242] [INSPIRE].ADSGoogle Scholar
  10. [10]
    P. Salmi and P. Sutcliffe, Aloof Baby Skyrmions, J. Phys. A 48 (2015) 035401 [arXiv:1409.8176] [INSPIRE].zbMATHADSGoogle Scholar
  11. [11]
    J. Ashcroft, M. Haberichter and S. Krusch, Baby Skyrme models without a potential term, Phys. Rev. D 91 (2015) 105032 [arXiv:1504.02459] [INSPIRE].MathSciNetADSGoogle Scholar
  12. [12]
    C. Adam, T. Romanczukiewicz, J. Sanchez-Guillen and A. Wereszczynski, Investigation of restricted baby Skyrme models, Phys. Rev. D 81 (2010) 085007 [arXiv:1002.0851] [INSPIRE].ADSGoogle Scholar
  13. [13]
    T. Gisiger and M.B. Paranjape, Solitons in a baby Skyrme model with invariance under volume/area preserving diffeomorphisms, Phys. Rev. D 55 (1997) 7731 [hep-ph/9606328] [INSPIRE].
  14. [14]
    A.N. Leznov, B. Piette and W.J. Zakrzewski, On the integrability of pure Skyrme models in two-dimensions, J. Math. Phys. 38 (1997) 3007 [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  15. [15]
    S. Bolognesi and W. Zakrzewski, Baby Skyrme Model, Near-BPS Approximations and Supersymmetric Extensions, Phys. Rev. D 91 (2015) 045034 [arXiv:1407.3140] [INSPIRE].ADSGoogle Scholar
  16. [16]
    S. Bolognesi and P. Sutcliffe, A low-dimensional analogue of holographic baryons, J. Phys. A 47 (2014) 135401 [arXiv:1311.2685] [INSPIRE].MathSciNetzbMATHADSGoogle Scholar
  17. [17]
    M. Elliot-Ripley and T. Winyard, Baby Skyrmions in AdS, JHEP 09 (2015) 009 [arXiv:1507.05928] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    M. Elliot-Ripley, Phases and approximations of baryonic popcorn in a low-dimensional analogue of holographic QCD, J. Phys. A 48 (2015) 295402 [arXiv:1503.08755] [INSPIRE].zbMATHGoogle Scholar
  19. [19]
    M. Elliot-Ripley, Salty popcorn in a homogeneous low-dimensional toy model of holographic QCD, J. Phys. A 50 (2017) 145401 [arXiv:1610.09169] [INSPIRE].MathSciNetzbMATHADSGoogle Scholar
  20. [20]
    T. Sakai and S. Sugimoto, Low energy hadron physics in holographic QCD, Prog. Theor. Phys. 113 (2005) 843 [hep-th/0412141] [INSPIRE].CrossRefzbMATHADSGoogle Scholar
  21. [21]
    S. Bolognesi and P. Sutcliffe, The Sakai-Sugimoto soliton, JHEP 01 (2014) 078 [arXiv:1309.1396] [INSPIRE].CrossRefADSGoogle Scholar
  22. [22]
    S. Baldino, S. Bolognesi, S.B. Gudnason and D. Koksal, Solitonic approach to holographic nuclear physics, Phys. Rev. D 96 (2017) 034008 [arXiv:1703.08695] [INSPIRE].ADSGoogle Scholar
  23. [23]
    L. Bartolini, S. Bolognesi and A. Proto, From the Sakai-Sugimoto Model to the Generalized Skyrme Model, Phys. Rev. D 97 (2018) 014024 [arXiv:1711.03873] [INSPIRE].ADSGoogle Scholar
  24. [24]
    G. Tallarita and F. Canfora, Multi-Skyrmions on AdS 2 × S 2 , Rational maps and Popcorn Transitions, Nucl. Phys. B 921 (2017) 394 [arXiv:1706.01397] [INSPIRE].zbMATHADSGoogle Scholar
  25. [25]
    I. Perapechka and Y. Shnir, Generalized Skyrmions and hairy black holes in asymptotically AdS 4 spacetime, Phys. Rev. D 95 (2017) 025024 [arXiv:1612.01914] [INSPIRE].ADSGoogle Scholar
  26. [26]
    V. Kaplunovsky, D. Melnikov and J. Sonnenschein, Baryonic Popcorn, JHEP 11 (2012) 047 [arXiv:1201.1331] [INSPIRE].CrossRefADSGoogle Scholar
  27. [27]
    M. Rho, S.-J. Sin and I. Zahed, Dense QCD: A Holographic Dyonic Salt, Phys. Lett. B 689 (2010) 23 [arXiv:0910.3774] [INSPIRE].CrossRefADSGoogle Scholar
  28. [28]
    P. Bizon and A. Wasserman, A Note on the non-existence of σ-model solitons in the 2 + 1 dimensional AdS gravity, Phys. Rev. D 71 (2005) 108701 [gr-qc/0411001] [INSPIRE].
  29. [29]
    C. Adam, J. Sanchez-Guillen and A. Wereszczynski, A Skyrme-type proposal for baryonic matter, Phys. Lett. B 691 (2010) 105 [arXiv:1001.4544] [INSPIRE].CrossRefADSGoogle Scholar
  30. [30]
    C. Adam, J. Sanchez-Guillen and A. Wereszczynski, A BPS Skyrme model and baryons at large N c, Phys. Rev. D 82 (2010) 085015 [arXiv:1007.1567] [INSPIRE].ADSGoogle Scholar
  31. [31]
    K. Sakamoto and K. Shiraishi, Boson stars with large selfinteraction in (2 + 1)-dimensions: An Exact solution, JHEP 07 (1998) 015 [gr-qc/9804067] [INSPIRE].
  32. [32]
    D. Astefanesei and E. Radu, Boson stars with negative cosmological constant, Nucl. Phys. B 665 (2003) 594 [gr-qc/0309131].
  33. [33]
    D. Astefanesei and E. Radu, Rotating boson stars in 2 + 1 dimensions, Phys. Lett. B 587 (2004) 7.MathSciNetCrossRefzbMATHADSGoogle Scholar
  34. [34]
    S.L. Liebling and C. Palenzuela, Dynamical Boson Stars, Living Rev. Rel. 15 (2012) 6.CrossRefzbMATHGoogle Scholar
  35. [35]
    G. Clement, Field-Theoretic Extended Particles in Two Space Dimensions, Nucl. Phys. B 114 (1976) 437 [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  36. [36]
    S.H. Mazharimousavi and M. Halilsoy, A topological metric in 2 + 1-dimensions, Eur. Phys. J. C 75 (2015) 249 [arXiv:1502.07662] [INSPIRE].CrossRefADSGoogle Scholar
  37. [37]
    B. Harms and A. Stern, Spinning σ-model solitons in 2 + 1 anti-de Sitter space, Phys. Lett. B 763 (2016) 401.CrossRefzbMATHADSGoogle Scholar
  38. [38]
    B. Harms and A. Stern, Growing Hair on the extremal BTZ black hole, Phys. Lett. B 769 (2017) 465 [arXiv:1703.10234] [INSPIRE].
  39. [39]
    C. Adam, C. Naya, J. Sanchez-Guillen, R. Vazquez and A. Wereszczynski, BPS Skyrmions as neutron stars, Phys. Lett. B 742 (2015) 136 [arXiv:1407.3799] [INSPIRE].CrossRefADSGoogle Scholar
  40. [40]
    C. Adam, C. Naya, J. Sanchez-Guillen, R. Vazquez and A. Wereszczynski, Neutron stars in the Bogomol’nyi-Prasad-Sommerfield Skyrme model: Mean-field limit versus full field theory, Phys. Rev. C 92 (2015) 025802 [arXiv:1503.03095] [INSPIRE].ADSGoogle Scholar
  41. [41]
    S.B. Gudnason, M. Nitta and N. Sawado, Gravitating BPS Skyrmions, JHEP 12 (2015) 013 [arXiv:1510.08735] [INSPIRE].MathSciNetzbMATHADSGoogle Scholar
  42. [42]
    A.A. Garcia and C. Campuzano, All static circularly symmetric perfect fluid solutions of (2 + 1) gravity, Phys. Rev. D 67 (2003) 064014 [gr-qc/0211014] [INSPIRE].
  43. [43]
    A. Banerjee, F. Rahaman, K. Jotania, R. Sharma and M. Rahaman, Exact solutions in (2 + 1)-dimensional anti-de Sitter space-time admitting a linear or non-linear equation of state, Astrophys. Space Sci. 355 (2015) 353 [arXiv:1412.3317] [INSPIRE].CrossRefADSGoogle Scholar
  44. [44]
    F. Cooper, H. Shepard and P. Sodano, Solitary waves in a class of generalized Korteweg-de Vries equations, Phys. Rev. E 48 (1993) 4027 [INSPIRE].MathSciNetADSGoogle Scholar
  45. [45]
    H. Arodz, Topological compactons, Acta Phys. Polon. B 33 (2002) 1241 [nlin/0201001] [INSPIRE].
  46. [46]
    H. Arodz, P. Klimas and T. Tyranowski, Field-theoretic models with V-shaped potentials, Acta Phys. Polon. B 36 (2005) 3861 [hep-th/0510204] [INSPIRE].zbMATHADSGoogle Scholar
  47. [47]
    D. Bazeia, E. da Hora, R. Menezes, H.P. de Oliveira and C. dos Santos, Compact-like kinks and vortices in generalized models, Phys. Rev. D 81 (2010) 125016 [arXiv:1004.3710] [INSPIRE].ADSGoogle Scholar
  48. [48]
    D. Bazeia, L. Losano and R. Menezes, New Results on Compact Structures, Phys. Lett. B 731 (2014) 293 [arXiv:1402.6617] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  49. [49]
    R. Casana, G. Lazar and L. Sourrouille, Self-dual effective compact and true compacton configurations in generalized Abelian Higgs models, Adv. High Energy Phys. 2018 (2018) 4281939 [arXiv:1709.01185] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  50. [50]
    J.M. Speight, Compactons and semi-compactons in the extreme baby Skyrme model, J. Phys. A 43 (2010) 405201 [arXiv:1006.3754] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  51. [51]
    B. Hartmann, B. Kleihaus, J. Kunz and I. Schaffer, Compact Boson Stars, Phys. Lett. B 714 (2012) 120 [arXiv:1205.0899] [INSPIRE].CrossRefADSGoogle Scholar
  52. [52]
    B. Hartmann, B. Kleihaus, J. Kunz and I. Schaffer, Compact (A)dS Boson Stars and Shells, Phys. Rev. D 88 (2013) 124033 [arXiv:1310.3632] [INSPIRE].ADSGoogle Scholar
  53. [53]
    B. Hartmann and J. Riedel, Supersymmetric Q-balls and boson stars in (d + 1) dimensions, Phys. Rev. D 87 (2013) 044003 [arXiv:1210.0096] [INSPIRE].ADSGoogle Scholar
  54. [54]
    S. Nelmes and B.M. A.G. Piette, Skyrmion stars and the multilayered rational map ansatz, Phys. Rev. D 84 (2011) 085017 [INSPIRE].ADSGoogle Scholar
  55. [55]
    S.G. Nelmes and B.M. A.G. Piette, Phase Transition and Anisotropic Deformations of Neutron Star Matter, Phys. Rev. D 85 (2012) 123004 [arXiv:1204.0910] [INSPIRE].ADSGoogle Scholar
  56. [56]
    C. Adam, C. Naya, J. Sanchez-Guillen, J.M. Speight and A. Wereszczynski, Thermodynamics of the BPS Skyrme model, Phys. Rev. D 90 (2014) 045003 [arXiv:1405.2927] [INSPIRE].ADSGoogle Scholar
  57. [57]
    C. Adam, J.M. Queiruga, J. Sanchez-Guillen and A. Wereszczynski, Extended Supersymmetry and BPS solutions in baby Skyrme models, JHEP 05 (2013) 108 [arXiv:1304.0774] [INSPIRE].MathSciNetCrossRefzbMATHADSGoogle Scholar
  58. [58]
    M. Nitta and S. Sasaki, BPS States in Supersymmetric Chiral Models with Higher Derivative Terms, Phys. Rev. D 90 (2014) 105001 [arXiv:1406.7647] [INSPIRE].ADSGoogle Scholar
  59. [59]
    M. Nitta and S. Sasaki, Classifying BPS States in Supersymmetric Gauge Theories Coupled to Higher Derivative Chiral Models, Phys. Rev. D 91 (2015) 125025 [arXiv:1504.08123] [INSPIRE].MathSciNetADSGoogle Scholar
  60. [60]
    J.M. Queiruga, Baby Skyrme model and fermionic zero modes, Phys. Rev. D 94 (2016) 065022 [arXiv:1606.02869] [INSPIRE].MathSciNetADSGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • C. Adam
    • 1
    Email author
  • T. Romanczukiewicz
    • 2
  • M. Wachla
    • 3
  • A. Wereszczynski
    • 2
  1. 1.Departamento de Física de PartículasUniversidad de Santiago de Compostela and Instituto Galego de Física de Altas Enerxias (IGFAE)Santiago de CompostelaSpain
  2. 2.Institute of PhysicsJagiellonian UniversityKrakówPoland
  3. 3.Institute of Nuclear PhysicsPolish Academy of SciencesKrakówPoland

Personalised recommendations