Journal of High Energy Physics

, 2014:143 | Cite as

Holographical description of BPS Wilson loops in flavored ABJM theory

Open Access
Regular Article - Theoretical Physics


As holographic description of BPS Wilson loops in \( \mathcal{N}=3 \) flavored ABJM theory with N f = k = 1, BPS M2-branes in AdS 4 × N (1, 1) are studied in details. Two 1/3-BPS membrane configurations are found. One of them is dual to the 1/3-BPS Wilson loop of Gaiotto-Yin type. The regulated membrane action captures precisely the leading exponential behavior of the vacuum expectation values of 1/3-BPS Wilson loops in the strong coupling limit, which was computed before using supersymmetric localization technique. Moreover, there is no BPS membrane with more supersymmetries in the background, under quite natural assumption on the membrane worldvolume. This suggests that there is no Wilson loop preserving more than 1/3 supersymmetries in such flavored ABJM theory.


Wilson ’t Hooft and Polyakov loops AdS-CFT Correspondence ChernSimons Theories 1/N Expansion 


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]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  2. [2]
    S.-J. Rey and J.-T. Yee, Macroscopic strings as heavy quarks in large-N gauge theory and anti-de Sitter supergravity, Eur. Phys. J. C 22 (2001) 379 [hep-th/9803001] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  3. [3]
    J.M. Maldacena, Wilson loops in large-N field theories, Phys. Rev. Lett. 80 (1998) 4859 [hep-th/9803002] [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  4. [4]
    N. Drukker and B. Fiol, All-genus calculation of Wilson loops using D-branes, JHEP 02 (2005) 010 [hep-th/0501109] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  5. [5]
    S. Yamaguchi, Wilson loops of anti-symmetric representation and D5-branes, JHEP 05 (2006) 037 [hep-th/0603208] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    B. Fiol, A. Guijosa and J.F. Pedraza, Branes from Light: Embeddings and Energetics for Symmetric k-Quarks in \( \mathcal{N}=4 \) SYM, arXiv:1410.0692 [INSPIRE].
  7. [7]
    E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].ADSCrossRefMATHMathSciNetGoogle Scholar
  8. [8]
    D. Gaiotto and X. Yin, Notes on superconformal Chern-Simons-Matter theories, JHEP 08 (2007) 056 [arXiv:0704.3740] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  9. [9]
    O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  10. [10]
    N. Drukker, J. Plefka and D. Young, Wilson loops in 3-dimensional N = 6 supersymmetric Chern-Simons Theory and their string theory duals, JHEP 11 (2008) 019 [arXiv:0809.2787] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  11. [11]
    B. Chen and J.-B. Wu, Supersymmetric Wilson Loops in N = 6 Super Chern-Simons-matter theory, Nucl. Phys. B 825 (2010) 38 [arXiv:0809.2863] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    S.-J. Rey, T. Suyama and S. Yamaguchi, Wilson Loops in Superconformal Chern-Simons Theory and Fundamental Strings in Anti-de Sitter Supergravity Dual, JHEP 03 (2009) 127 [arXiv:0809.3786] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  13. [13]
    N. Drukker and D. Trancanelli, A Supermatrix model for N = 6 super Chern-Simons-matter theory, JHEP 02 (2010) 058 [arXiv:0912.3006] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  14. [14]
    A. Kapustin, B. Willett and I. Yaakov, Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  15. [15]
    M. Mariño and P. Putrov, Exact Results in ABJM Theory from Topological Strings, JHEP 06 (2010) 011 [arXiv:0912.3074] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    K. Hosomichi, K.-M. Lee, S. Lee, S. Lee and J. Park, N=5,6 Superconformal Chern-Simons Theories and M2-branes on Orbifolds, JHEP 09 (2008) 002 [arXiv:0806.4977] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  17. [17]
    O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP 11 (2008) 043 [arXiv:0807.4924] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  18. [18]
    K.-M. Lee and S. Lee, 1/2-BPS Wilson Loops and Vortices in ABJM Model, JHEP 09 (2010) 004 [arXiv:1006.5589] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    H. Kim, N. Kim and J.H. Lee, One-loop corrections to holographic Wilson loop in AdS4xCP3, J. Korean Phys. Soc. 61 (2012) 713 [arXiv:1203.6343] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    A. Klemm, M. Mariño, M. Schiereck and M. Soroush, ABJM Wilson loops in the Fermi gas approach, arXiv:1207.0611 [INSPIRE].
  21. [21]
    L. Griguolo, D. Marmiroli, G. Martelloni and D. Seminara, The generalized cusp in ABJ(M) \( N\mathcal{N}=6 \) Super Chern-Simons theories, JHEP 05 (2013) 113 [arXiv:1208.5766] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  22. [22]
    V. Cardinali, L. Griguolo, G. Martelloni and D. Seminara, New supersymmetric Wilson loops in ABJ(M) theories, Phys. Lett. B 718 (2012) 615 [arXiv:1209.4032] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  23. [23]
    M.S. Bianchi, G. Giribet, M. Leoni and S. Penati, 1/2 BPS Wilson loop in N = 6 superconformal Chern-Simons theory at two loops, Phys. Rev. D 88 (2013) 026009 [arXiv:1303.6939] [INSPIRE].ADSGoogle Scholar
  24. [24]
    D. Farquet and J. Sparks, Wilson loops and the geometry of matrix models in AdS 4 /CFT 3, JHEP 01 (2014) 083 [arXiv:1304.0784] [INSPIRE].CrossRefGoogle Scholar
  25. [25]
    T. Suyama, A Systematic Study on Matrix Models for Chern-Simons-matter Theories, Nucl. Phys. B 874 (2013) 528 [arXiv:1304.7831] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  26. [26]
    A. Grassi, J. Kallen and M. Mariño, The topological open string wavefunction, arXiv:1304.6097 [INSPIRE].
  27. [27]
    N. Kim, Supersymmetric Wilson loops with general contours in ABJM theory, Mod. Phys. Lett. A 28 (2013) 1350150 [arXiv:1304.7660] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    Y. Hatsuda, M. Honda, S. Moriyama and K. Okuyama, ABJM Wilson Loops in Arbitrary Representations, JHEP 10 (2013) 168 [arXiv:1306.4297] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    M.S. Bianchi, G. Giribet, M. Leoni and S. Penati, The 1/2 BPS Wilson loop in ABJ(M) at two loops: The details, JHEP 10 (2013) 085 [arXiv:1307.0786] [INSPIRE].ADSCrossRefGoogle Scholar
  30. [30]
    L. Griguolo, G. Martelloni, M. Poggi and D. Seminara, Perturbative evaluation of circular 1/2 BPS Wilson loops in \( \mathcal{N}=6 \) Super Chern-Simons theories, JHEP 09 (2013) 157 [arXiv:1307.0787] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  31. [31]
    J.-B. Wu and M.-Q. Zhu, BPS M2-branes in AdS 4 × Q 1,1,1 and their dual loop operators, Phys. Rev. D 89 (2014) 126003 [arXiv:1312.3030] [INSPIRE].ADSGoogle Scholar
  32. [32]
    M.S. Bianchi, L. Griguolo, M. Leoni, S. Penati and D. Seminara, BPS Wilson loops and Bremsstrahlung function in ABJ(M): a two loop analysis, JHEP 06 (2014) 123 [arXiv:1402.4128] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  33. [33]
    D.H. Correa, J. Aguilera-Damia and G.A. Silva, Strings in \( Ad{S}_4 \times \mathbb{C}{\mathrm{\mathbb{P}}}^3 \) Wilson loops in \( \mathcal{N}=6 \) super Chern-Simons-matter and bremsstrahlung functions, JHEP 06 (2014) 139 [arXiv:1405.1396] [INSPIRE].ADSMathSciNetGoogle Scholar
  34. [34]
    D. Farquet and J. Sparks, Wilson loops on three-manifolds and their M2-brane duals, arXiv:1406.2493 [INSPIRE].
  35. [35]
    S. Hohenegger and I. Kirsch, A Note on the holography of Chern-Simons matter theories with flavour, JHEP 04 (2009) 129 [arXiv:0903.1730] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    D. Gaiotto and D.L. Jafferis, Notes on adding D6 branes wrapping Rp 3 in AdS 4 × CP 3, JHEP 11 (2012) 015 [arXiv:0903.2175] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  37. [37]
    Y. Hikida, W. Li and T. Takayanagi, ABJM with Flavors and FQHE, JHEP 07 (2009) 065 [arXiv:0903.2194] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  38. [38]
    E. Conde and A.V. Ramallo, On the gravity dual of Chern-Simons-matter theories with unquenched flavor, JHEP 07 (2011) 099 [arXiv:1105.6045] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  39. [39]
    J.P. Gauntlett, G.W. Gibbons, G. Papadopoulos and P.K. Townsend, Hyper-Kähler manifolds and multiply intersecting branes, Nucl. Phys. B 500 (1997) 133 [hep-th/9702202] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  40. [40]
    D.N. Page and C.N. Pope, New Squashed Solutions of D = 11 Supergravity, Phys. Lett. B 147 (1984) 55 [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  41. [41]
    J.P. Gauntlett, S. Lee, T. Mateos and D. Waldram, Marginal deformations of field theories with AdS 4 duals, JHEP 08 (2005) 030 [hep-th/0505207] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  42. [42]
    M. Fujita and T.-S. Tai, Eschenburg space as gravity dual of flavored N = 4 Chern-Simons-matter theory, JHEP 09 (2009) 062 [arXiv:0906.0253] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  43. [43]
    R.C. Santamaria, M. Mariño and P. Putrov, Unquenched flavor and tropical geometry in strongly coupled Chern-Simons-matter theories, JHEP 10 (2011) 139 [arXiv:1011.6281] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    N. Drukker, D.J. Gross and H. Ooguri, Wilson loops and minimal surfaces, Phys. Rev. D 60 (1999) 125006 [hep-th/9904191] [INSPIRE].ADSMathSciNetGoogle Scholar
  45. [45]
    N. Drukker, 1/4 BPS circular loops, unstable world-sheet instantons and the matrix model, JHEP 09 (2006) 004 [hep-th/0605151] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  46. [46]
    N. Drukker, S. Giombi, R. Ricci and D. Trancanelli, On the D3-brane description of some 1/4 BPS Wilson loops, JHEP 04 (2007) 008 [hep-th/0612168] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  47. [47]
    K. Hosomichi, K.-M. Lee, S. Lee, S. Lee and J. Park, N=4 Superconformal Chern-Simons Theories with Hyper and Twisted Hyper Multiplets, JHEP 07 (2008) 091 [arXiv:0805.3662] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  48. [48]
    M. Benna, I. Klebanov, T. Klose and M. Smedback, Superconformal Chern-Simons Theories and AdS 4 /CFT 3 Correspondence, JHEP 09 (2008) 072 [arXiv:0806.1519] [INSPIRE].ADSCrossRefMathSciNetGoogle Scholar
  49. [49]
    Y. Imamura and K. Kimura, On the moduli space of elliptic Maxwell- Chern-Simons theories, Prog. Theor. Phys. 120 (2008) 509 [arXiv:0806.3727] [INSPIRE].ADSCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Department of Physics, and State Key Laboratory of Nuclear Physics and TechnologyPeking UniversityBeijingP. R. China
  2. 2.Collaborative Innovation Center of Quantum MatterBeijingP. R. China
  3. 3.Center for High Energy PhysicsPeking UniversityBeijingP. R. China
  4. 4.Institute of High Energy Physics, and Theoretical Physics Center for Science FacilitiesChinese Academy of SciencesBeijingP. R. China
  5. 5.SISSATriesteItaly

Personalised recommendations