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Journal of High Energy Physics

, 2011:17 | Cite as

Vortices and monopoles in mass-deformed SO and USp gauge theories

  • Minoru Eto
  • Toshiaki Fujimori
  • Sven Bjarke Gudnason
  • Yunguo Jiang
  • Kenichi Konishi
  • Muneto Nitta
  • Keisuke Ohashi
Article

Abstract

Effects of mass deformations on 1/2 Bogomol’nyi-Prasad-Sommerfield (BPS) non-Abelian vortices are studied in 4 d N = 2 supersymmetric U(1) × SO(2n) and U(1) × USp(2n) gauge theories, with Nf = 2n quark multiplets. The 2 d N = (2, 2) effective world-sheet sigma models on the Hermitian symmetric spaces SO(2n)/U(n) and USp(2n)/U(n) found recently which describe the low-energy excitations of the orientational moduli of the vortices, are generalized to the respective massive sigma models. The continuous vortex moduli spaces are replaced by a finite number (2n−1 or 2n) of vortex solutions. The 1/2 BPS kinks connecting different vortex vacua are magnetic monopoles in the 4 d theory, trapped inside the vortex core, with total configurations being 1/4 BPS composite states. These configurations are systematically studied within the semi-classical regime.

Keywords

Solitons Monopoles and Instantons Supersymmetric gauge theory Sigma Models 

References

  1. [1]
    A. Hanany and D. Tong, Vortices, instantons and branes, JHEP 07 (2003) 037 [hep-th/0306150] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  2. [2]
    R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi and A. Yung, NonAbelian superconductors: vortices and confinement in N = 2 SQCD, Nucl. Phys. B 673 (2003) 187 [hep-th/0307287] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  3. [3]
    D. Tong, Monopoles in the Higgs phase, Phys. Rev. D 69 (2004) 065003 [hep-th/0307302] [INSPIRE].ADSGoogle Scholar
  4. [4]
    R. Auzzi, S. Bolognesi, J. Evslin and K. Konishi, NonAbelian monopoles and the vortices that confine them, Nucl. Phys. B 686 (2004) 119 [hep-th/0312233] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  5. [5]
    A. Hanany and D. Tong, Vortex strings and four-dimensional gauge dynamics, JHEP 04 (2004) 066 [hep-th/0403158] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  6. [6]
    M. Shifman and A. Yung, NonAbelian string junctions as confined monopoles, Phys. Rev. D 70 (2004) 045004 [hep-th/0403149] [INSPIRE].ADSMathSciNetGoogle Scholar
  7. [7]
    M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, Instantons in the Higgs phase, Phys. Rev. D 72 (2005) 025011 [hep-th/0412048] [INSPIRE].ADSMathSciNetGoogle Scholar
  8. [8]
    A. Gorsky, M. Shifman and A. Yung, Non-Abelian Meissner effect in Yang-Mills theories at weak coupling, Phys. Rev. D 71 (2005) 045010 [hep-th/0412082] [INSPIRE].ADSGoogle Scholar
  9. [9]
    S.B. Gudnason, Y. Jiang and K. Konishi, Non-Abelian vortex dynamics: effective world-sheet action, JHEP 08 (2010) 012 [arXiv:1007.2116] [INSPIRE].CrossRefMathSciNetGoogle Scholar
  10. [10]
    M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, Moduli space of non-Abelian vortices, Phys. Rev. Lett. 96 (2006) 161601 [hep-th/0511088] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  11. [11]
    M. Eto et al., Non-Abelian vortices of higher winding numbers, Phys. Rev. D 74 (2006) 065021 [hep-th/0607070] [INSPIRE].ADSMathSciNetGoogle Scholar
  12. [12]
    M. Eto et al., Universal reconnection of non-Abelian cosmic strings, Phys. Rev. Lett. 98 (2007) 091602 [hep-th/0609214] [INSPIRE].CrossRefADSGoogle Scholar
  13. [13]
    T. Fujimori, G. Marmorini, M. Nitta, K. Ohashi and N. Sakai, The moduli space metric for well-separated non-Abelian vortices, Phys. Rev. D 82 (2010) 065005 [arXiv:1002.4580] [INSPIRE].ADSGoogle Scholar
  14. [14]
    M. Eto, T. Fujimori, M. Nitta, K. Ohashi and N. Sakai, Dynamics of non-Abelian vortices, arXiv:1105.1547 [INSPIRE].
  15. [15]
    M. Eto et al., Constructing non-Abelian vortices with arbitrary gauge groups, Phys. Lett. B 669 (2008) 98 [arXiv:0802.1020] [INSPIRE].ADSMathSciNetGoogle Scholar
  16. [16]
    M. Eto, T. Fujimori, S.B. Gudnason, M. Nitta and K. Ohashi, SO and USp Kähler and hyper-Kähler quotients and lumps, Nucl. Phys. B 815 (2009) 495 [arXiv:0809.2014] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  17. [17]
    M. Eto et al., Non-Abelian vortices in SO(N ) and USp(N ) gauge theories, JHEP 06 (2009) 004 [arXiv:0903.4471] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  18. [18]
    Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, Construction of non-Abelian walls and their complete moduli space, Phys. Rev. Lett. 93 (2004) 161601 [hep-th/0404198] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  19. [19]
    Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, Non-Abelian walls in supersymmetric gauge theories, Phys. Rev. D 70 (2004) 125014 [hep-th/0405194] [INSPIRE].ADSMathSciNetGoogle Scholar
  20. [20]
    Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, All exact solutions of a 1/4 Bogomol’nyi-Prasad-Sommerfield equation, Phys. Rev. D 71 (2005) 065018 [hep-th/0405129] [INSPIRE].ADSMathSciNetGoogle Scholar
  21. [21]
    M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, Solitons in the Higgs phase: the moduli matrix approach, J. Phys. A A 39 (2006) R315 [hep-th/0602170] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  22. [22]
    N. Dorey, The BPS spectra of two-dimensional supersymmetric gauge theories with twisted mass terms, JHEP 11 (1998) 005 [hep-th/9806056] [INSPIRE].ADSGoogle Scholar
  23. [23]
    N. Dorey, T.J. Hollowood and D. Tong, The BPS spectra of gauge theories in two-dimensions and four-dimensions, JHEP 05 (1999) 006 [hep-th/9902134] [INSPIRE].CrossRefADSGoogle Scholar
  24. [24]
    G. Carlino, K. Konishi and H. Murayama, Dynamics of supersymmetric SU(n c) and USp(2n c) gauge theories, JHEP 02 (2000) 004 [hep-th/0001036] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  25. [25]
    G. Carlino, K. Konishi and H. Murayama, Dynamical symmetry breaking in supersymmetric SU(n c) and USp(2n c) gauge theories, Nucl. Phys. B 590 (2000) 37 [hep-th/0005076] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  26. [26]
    S. Bolognesi, K. Konishi and G. Marmorini, Light nonAbelian monopoles and generalized r-vacua in supersymmetric gauge theories, Nucl. Phys. B 718 (2005) 134 [hep-th/0502004] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  27. [27]
    D. Dorigoni, K. Konishi and K. Ohashi, Non-Abelian vortices with product moduli, Phys. Rev. D 79 (2009) 045011 [arXiv:0801.3284] [INSPIRE].ADSGoogle Scholar
  28. [28]
    M. Eto et al., Group theory of non-Abelian vortices, JHEP 11 (2010) 042 [arXiv:1009.4794] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  29. [29]
    J. Scherk and J.H. Schwarz, Spontaneous breaking of supersymmetry through dimensional reduction, Phys. Lett. B 82 (1979) 60 [INSPIRE].ADSGoogle Scholar
  30. [30]
    J. Scherk and J.H. Schwarz, How to get masses from extra dimensions, Nucl. Phys. B 153 (1979)61 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  31. [31]
    N. Sakai and D. Tong, Monopoles, vortices, domain walls and D-branes: the rules of interaction, JHEP 03 (2005) 019 [hep-th/0501207] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  32. [32]
    M. Arai and S. Shin, Walls of massive Kähler σ-models on SO(2N )/U(N ) and Sp(N )/U(N ), Phys. Rev. D 83 (2011) 125003 [arXiv:1103.1490] [INSPIRE].ADSGoogle Scholar
  33. [33]
    M. Eto et al., Dynamics of strings between walls, Phys. Rev. D 79 (2009) 045015 [arXiv:0810.3495] [INSPIRE].ADSGoogle Scholar
  34. [34]
    M. Eto et al., Fractional vortices and lumps, Phys. Rev. D 80 (2009) 045018 [arXiv:0905.3540] [INSPIRE].ADSGoogle Scholar
  35. [35]
    F. Delduc and G. Valent, Classical and quantum structure of the compact Kählerian σ-models, Nucl. Phys. B 253 (1985) 494 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  36. [36]
    F. Delduc and G. Valent, Renormalizability of the generalized σ-models defined on compact Hermitian symmetric spaces, Phys. Lett. B 148 (1984) 124 [INSPIRE].ADSMathSciNetGoogle Scholar
  37. [37]
    M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, Manifestly supersymmetric effective Lagrangians on BPS solitons, Phys. Rev. D 73 (2006) 125008 [hep-th/0602289] [INSPIRE].ADSMathSciNetGoogle Scholar
  38. [38]
    M. Shifman and A. Yung, Localization of nonAbelian gauge fields on domain walls at weak coupling (D-brane prototypes II), Phys. Rev. D 70 (2004) 025013 [hep-th/0312257] [INSPIRE].ADSMathSciNetGoogle Scholar
  39. [39]
    M. Eto, M. Nitta, K. Ohashi and D. Tong, Skyrmions from instantons inside domain walls, Phys. Rev. Lett. 95 (2005) 252003 [hep-th/0508130] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  40. [40]
    M. Eto, T. Fujimori, M. Nitta, K. Ohashi and N. Sakai, Domain walls with non-Abelian clouds, Phys. Rev. D 77 (2008) 125008 [arXiv:0802.3135] [INSPIRE].ADSGoogle Scholar
  41. [41]
    M. Nitta and W. Vinci, Non-Abelian monopoles in the Higgs phase, Nucl. Phys. B 848 (2011) 121 [arXiv:1012.4057] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  42. [42]
    L. Álvarez-Gaumé and D.Z. Freedman, Potentials for the supersymmetric nonlinear σ-model, Commun. Math. Phys. 91 (1983) 87 [INSPIRE].CrossRefADSGoogle Scholar
  43. [43]
    S.J.J. Gates, Superspace formulation of new nonlinear σ-models, Nucl. Phys. B 238 (1984) 349 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  44. [44]
    D. Tong, A note on 1/4 BPS states, Phys. Lett. B 460 (1999) 295 [hep-th/9902005] [INSPIRE].ADSGoogle Scholar
  45. [45]
    E.R.C. Abraham and P.K. Townsend, Q kinks, Phys. Lett. B 291 (1992) 85 [INSPIRE].ADSMathSciNetGoogle Scholar
  46. [46]
    E.R.C. Abraham and P.K. Townsend, More on Q kinks: a (1+1)-dimensional analog of dyons, Phys. Lett. B 295 (1992) 225 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  47. [47]
    M. Arai, M. Naganuma, M. Nitta and N. Sakai, Manifest supersymmetry for BPS walls in N =2 nonlinear σ-models, Nucl. Phys. B 652(2003)35 [hep-th/0211103] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  48. [48]
    J.P. Gauntlett, D. Tong and P.K. Townsend, Multidomain walls in massive supersymmetric σ-models, Phys. Rev. D 64 (2001) 025010 [hep-th/0012178] [INSPIRE].ADSMathSciNetGoogle Scholar
  49. [49]
    D. Tong, The moduli space of BPS domain walls, Phys. Rev. D 66 (2002) 025013 [hep-th/0202012] [INSPIRE].ADSGoogle Scholar
  50. [50]
    E. Witten, Dynamical breaking of supersymmetry, Nucl. Phys. B 188 (1981) 513 [INSPIRE].CrossRefADSGoogle Scholar
  51. [51]
    E. Witten, Constraints on supersymmetry breaking, Nucl. Phys. B 202 (1982) 253 [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  52. [52]
    K. Higashijima and M. Nitta, Supersymmetric nonlinear σ-models as gauge theories, Prog. Theor. Phys. 103 (2000) 635 [hep-th/9911139] [INSPIRE].CrossRefzbMATHADSMathSciNetGoogle Scholar
  53. [53]
    M. Eto et al., D-brane construction for non-Abelian walls, Phys. Rev. D 71 (2005) 125006 [hep-th/0412024] [INSPIRE].ADSMathSciNetGoogle Scholar
  54. [54]
    M. Arai, M. Nitta and N. Sakai, Vacua of massive hyperKähler σ-models of nonAbelian quotient, Prog. Theor. Phys. 113 (2005) 657 [hep-th/0307274] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  55. [55]
    M. Eto et al., Statistical mechanics of vortices from D-branes and T-duality, Nucl. Phys. B 788 (2008) 120 [hep-th/0703197] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  56. [56]
    E. Witten, Dyons of charge eθ/2π, Phys. Lett. B 86 (1979) 283 [INSPIRE].ADSGoogle Scholar
  57. [57]
    K. Hashimoto, H. Hata and N. Sasakura, Three-string junction and BPS saturated solutions in SU(3) supersymmetric Yang-Mills theory, Phys. Lett. B 431 (1998) 303 [hep-th/9803127] [INSPIRE].ADSMathSciNetGoogle Scholar
  58. [58]
    K. Hashimoto, H. Hata and N. Sasakura, Multipronged strings and BPS saturated solutions in SU(N ) supersymmetric Yang-Mills theory, Nucl. Phys. B 535 (1998) 83 [hep-th/9804164] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  59. [59]
    K.-M. Lee and P. Yi, Dyons in N = 4 supersymmetric theories and three pronged strings, Phys. Rev. D 58 (1998) 066005 [hep-th/9804174] [INSPIRE].ADSMathSciNetGoogle Scholar
  60. [60]
    M. Eto, Y. Isozumi, M. Nitta and K. Ohashi, 1/2, 1/4 and 1/8 BPS equations in SUSY Yang-Mills-Higgs systems: field theoretical brane configurations, Nucl. Phys. B 752 (2006) 140 [hep-th/0506257] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  61. [61]
    A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles and three-dimensional gauge dynamics, Nucl. Phys. B 492 (1997) 152 [hep-th/9611230] [INSPIRE].ADSMathSciNetGoogle Scholar
  62. [62]
    N.J. Evans, C.V. Johnson and A.D. Shapere, Orientifolds, branes and duality of 4-D gauge theories, Nucl. Phys. B 505 (1997) 251 [hep-th/9703210] [INSPIRE].CrossRefADSMathSciNetGoogle Scholar
  63. [63]
    M. Shifman and A. Yung, Non-Abelian semilocal strings in N = 2 supersymmetric QCD, Phys. Rev. D 73 (2006) 125012 [hep-th/0603134] [INSPIRE].ADSGoogle Scholar
  64. [64]
    M. Eto et al., On the moduli space of semilocal strings and lumps, Phys. Rev. D 76 (2007) 105002 [arXiv:0704.2218] [INSPIRE].ADSMathSciNetGoogle Scholar
  65. [65]
    M. Shifman, W. Vinci and A. Yung, Effective world-sheet theory for non-Abelian semilocal strings in N = 2 supersymmetric QCD, Phys. Rev. D 83 (2011) 125017 [arXiv:1104.2077] [INSPIRE].ADSGoogle Scholar
  66. [66]
    K. Higashijima, T. Kimura, M. Nitta and M. Tsuzuki, Large-N limit of N = 2 supersymmetric Q N model in two-dimensions, Prog. Theor. Phys. 105 (2001) 261 [hep-th/0010272] [INSPIRE].CrossRefzbMATHADSMathSciNetGoogle Scholar
  67. [67]
    M. Arai, S. Lee and S. Shin, Walls in supersymmetric massive nonlinear σ-model on complex quadric surface, Phys. Rev. D 80 (2009) 125012 [arXiv:0908.3713] [INSPIRE].ADSGoogle Scholar
  68. [68]
    N.S. Manton and P. Sutcliffe, Topological solitons, Cambridge University Press, Cambridge U.K. (2004).CrossRefzbMATHGoogle Scholar
  69. [69]
    R.S. Palais, The principle of symmetric criticality, Commun. Math. Phys. 69 (1979) 19.CrossRefzbMATHADSMathSciNetGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2011

Authors and Affiliations

  • Minoru Eto
    • 1
  • Toshiaki Fujimori
    • 2
    • 3
  • Sven Bjarke Gudnason
    • 4
  • Yunguo Jiang
    • 5
    • 6
  • Kenichi Konishi
    • 2
    • 3
  • Muneto Nitta
    • 7
  • Keisuke Ohashi
    • 8
  1. 1.Department of PhysicsYamagata UniversityYamagataJapan
  2. 2.INFN, Sezione di PisaPisaItaly
  3. 3.Department of Physics, “E. Fermi”University of PisaPisaItaly
  4. 4.Racah Institute of PhysicsThe Hebrew UniversityJerusalemIsrael
  5. 5.Institute of High Energy PhysicsChinese Academy of SciencesBeijingChina
  6. 6.Theoretical Physics Center for Science FacilitiesChinese Academy of SciencesBeijingChina
  7. 7.Department of Physics, and Research and Education Center for Natural SciencesKeio UniversityYokohamaJapan
  8. 8.Department of PhysicsKyoto UniversityKyotoJapan

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