A quantum check of non-supersymmetric AdS/dCFT

  • Aleix Gimenez Grau
  • Charlotte Kristjansen
  • Matthias VolkEmail author
  • Matthias Wilhelm
Open Access
Regular Article - Theoretical Physics


Via a challenging field-theory computation, we confirm a supergravity prediction for the non-supersymmetric D3-D7 probe-brane system with probe geometry AdS4 ×S2 ×S2, stabilized by fluxes. Supergravity predicts, in a certain double-scaling limit, the value of the one-point functions of chiral primaries of the dual defect version of \( \mathcal{N}=4 \) SYM theory, where the fluxes translate into SO(3) × SO(3)-symmetric, Lie-algebra-valued vacuum expectation values for all six scalar fields. Using a generalization of the technique based on fuzzy spherical harmonics developed for the related D3-D5 probe-brane system, we diagonalize the resulting mass matrix of the field theory. Subsequently, we calculate the planar one-loop correction to the vacuum expectation values of the scalars in dimensional reduction and find that it is UV finite and non-vanishing. We then proceed to calculating the one-loop correction to the planar one-point function of any single-trace scalar operator and explicitly evaluate this correction for a 1/2-BPS operator of length L at two leading orders in the double-scaling limit, finding exact agreement with the supergravity prediction.


1/N Expansion AdS-CFT Correspondence Supersymmetric Gauge Theory 


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]
    N. Andrei et al., Boundary and Defect CFT: Open Problems and Applications, 2018, arXiv:1810.05697 [INSPIRE].
  2. [2]
    N.R. Constable, R.C. Myers and O. Tafjord, The Noncommutative bion core, Phys. Rev. D 61 (2000) 106009 [hep-th/9911136] [INSPIRE].
  3. [3]
    A. Karch and L. Randall, Open and closed string interpretation of SUSY CFT’s on branes with boundaries, JHEP 06 (2001) 063 [hep-th/0105132] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    M. de Leeuw, A.C. Ipsen, C. Kristjansen and M. Wilhelm, Introduction to Integrability and One-point Functions in \( \mathcal{N}=4 \) SYM and its Defect Cousin, in Les Houches Summer School: Integrability: From Statistical Systems to Gauge Theory, Les Houches France (2016) [arXiv:1708.02525] [INSPIRE].
  5. [5]
    O. DeWolfe, D.Z. Freedman and H. Ooguri, Holography and defect conformal field theories, Phys. Rev. D 66 (2002) 025009 [hep-th/0111135] [INSPIRE].
  6. [6]
    J. Erdmenger, Z. Guralnik and I. Kirsch, Four-dimensional superconformal theories with interacting boundaries or defects, Phys. Rev. D 66 (2002) 025020 [hep-th/0203020] [INSPIRE].
  7. [7]
    J.L. Cardy, Conformal Invariance and Surface Critical Behavior, Nucl. Phys. B 240 (1984) 514 [INSPIRE].
  8. [8]
    M. de Leeuw, C. Kristjansen and K. Zarembo, One-point Functions in Defect CFT and Integrability, JHEP 08 (2015) 098 [arXiv:1506.06958] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    I. Buhl-Mortensen, M. de Leeuw, C. Kristjansen and K. Zarembo, One-point Functions in AdS/dCFT from Matrix Product States, JHEP 02 (2016) 052 [arXiv:1512.02532] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    M. de Leeuw, C. Kristjansen and S. Mori, AdS/dCFT one-point functions of the SU(3) sector, Phys. Lett. B 763 (2016) 197 [arXiv:1607.03123] [INSPIRE].
  11. [11]
    M. De Leeuw, C. Kristjansen and G. Linardopoulos, Scalar one-point functions and matrix product states of AdS/dCFT, Phys. Lett. B 781 (2018) 238 [arXiv:1802.01598] [INSPIRE].
  12. [12]
    I. Buhl-Mortensen, M. de Leeuw, A.C. Ipsen, C. Kristjansen and M. Wilhelm, One-loop one-point functions in gauge-gravity dualities with defects, Phys. Rev. Lett. 117 (2016) 231603 [arXiv:1606.01886] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    I. Buhl-Mortensen, M. de Leeuw, A.C. Ipsen, C. Kristjansen and M. Wilhelm, A Quantum Check of AdS/dCFT, JHEP 01 (2017) 098 [arXiv:1611.04603] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    I. Buhl-Mortensen, M. de Leeuw, A.C. Ipsen, C. Kristjansen and M. Wilhelm, Asymptotic One-Point Functions in Gauge-String Duality with Defects, Phys. Rev. Lett. 119 (2017) 261604 [arXiv:1704.07386] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    K. Nagasaki and S. Yamaguchi, Expectation values of chiral primary operators in holographic interface CFT, Phys. Rev. D 86 (2012) 086004 [arXiv:1205.1674] [INSPIRE].
  16. [16]
    K. Nagasaki, H. Tanida and S. Yamaguchi, Holographic Interface-Particle Potential, JHEP 01 (2012) 139 [arXiv:1109.1927] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  17. [17]
    D.E. Berenstein, J.M. Maldacena and H.S. Nastase, Strings in flat space and pp waves from \( \mathcal{N}=4 \) superYang-Mills, JHEP 04 (2002) 013 [hep-th/0202021] [INSPIRE].
  18. [18]
    Z. Bern, M. Czakon, L.J. Dixon, D.A. Kosower and V.A. Smirnov, The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory, Phys. Rev. D 75 (2007) 085010 [hep-th/0610248] [INSPIRE].
  19. [19]
    N. Beisert, B. Eden and M. Staudacher, Transcendentality and Crossing, J. Stat. Mech. 0701 (2007) P01021 [hep-th/0610251] [INSPIRE].
  20. [20]
    F. Cachazo, M. Spradlin and A. Volovich, Four-loop cusp anomalous dimension from obstructions, Phys. Rev. D 75 (2007) 105011 [hep-th/0612309] [INSPIRE].
  21. [21]
    C. Kristjansen, G.W. Semenoff and D. Young, Chiral primary one-point functions in the D3-D7 defect conformal field theory, JHEP 01 (2013) 117 [arXiv:1210.7015] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    O. Bergman, N. Jokela, G. Lifschytz and M. Lippert, Quantum Hall Effect in a Holographic Model, JHEP 10 (2010) 063 [arXiv:1003.4965] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  23. [23]
    R.C. Myers and M.C. Wapler, Transport Properties of Holographic Defects, JHEP 12 (2008) 115 [arXiv:0811.0480] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    G. Grignani, N. Kim and G.W. Semenoff, D3-D5 holography with flux, Phys. Lett. B 715 (2012) 225 [arXiv:1203.6162] [INSPIRE].
  25. [25]
    C. Kristjansen and G.W. Semenoff, Giant D5 Brane Holographic Hall State, JHEP 06 (2013) 048 [arXiv:1212.5609] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    C. Kristjansen, R. Pourhasan and G.W. Semenoff, A Holographic Quantum Hall Ferromagnet, JHEP 02 (2014) 097 [arXiv:1311.6999] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    J. Hutchinson, C. Kristjansen and G.W. Semenoff, Conductivity Tensor in a Holographic Quantum Hall Ferromagnet, Phys. Lett. B 738 (2014) 373 [arXiv:1408.3320] [INSPIRE].
  28. [28]
    N. Jokela, G. Lifschytz and M. Lippert, Holographic anyonic superfluidity, JHEP 10 (2013) 014 [arXiv:1307.6336] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    N. Jokela, G. Lifschytz and M. Lippert, Flowing holographic anyonic superfluid, JHEP 10 (2014) 21 [arXiv:1407.3794] [INSPIRE].
  30. [30]
    A. Gimenez Grau, C. Kristjansen, M. Volk and M. Wilhelm, work in progress.Google Scholar
  31. [31]
    A. Chatzistavrakidis, H. Steinacker and G. Zoupanos, On the fermion spectrum of spontaneously generated fuzzy extra dimensions with fluxes, Fortsch. Phys. 58 (2010) 537 [arXiv:0909.5559] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    L.F. Alday, J.M. Henn, J. Plefka and T. Schuster, Scattering into the fifth dimension of \( \mathcal{N}=4 \) super Yang-Mills, JHEP 01 (2010) 077 [arXiv:0908.0684] [INSPIRE].
  33. [33]
    C.P. Burgess and G.D. Moore, The standard model: A primer, Cambridge University Press, Cambridge U.K. (2006).Google Scholar
  34. [34]
    M. Ammon and J. Erdmenger, Gauge/gravity duality, Cambridge University Press, Cambridge U.K. (2015).Google Scholar
  35. [35]
    T. Kawano and K. Okuyama, Spinor exchange in AdS(d + 1), Nucl. Phys. B 565 (2000) 427 [hep-th/9905130] [INSPIRE].
  36. [36]
    D. Gaiotto and E. Witten, Supersymmetric Boundary Conditions in \( \mathcal{N}=4 \) Super Yang-Mills Theory, J. Statist. Phys. 135 (2009) 789 [arXiv:0804.2902] [INSPIRE].
  37. [37]
    M. de Leeuw, A.C. Ipsen, C. Kristjansen, K.E. Vardinghus and M. Wilhelm, Two-point functions in AdS/dCFT and the boundary conformal bootstrap equations, JHEP 08 (2017) 020 [arXiv:1705.03898] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    W. Siegel, Supersymmetric Dimensional Regularization via Dimensional Reduction, Phys. Lett. 84B (1979) 193 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    D.M. Capper, D.R.T. Jones and P. van Nieuwenhuizen, Regularization by Dimensional Reduction of Supersymmetric and Nonsupersymmetric Gauge Theories, Nucl. Phys. B 167 (1980) 479 [INSPIRE].
  40. [40]
    J.K. Erickson, G.W. Semenoff and K. Zarembo, Wilson loops in \( \mathcal{N}=4 \) supersymmetric Yang-Mills theory, Nucl. Phys. B 582 (2000) 155 [hep-th/0003055] [INSPIRE].
  41. [41]
    D. Nandan, C. Sieg, M. Wilhelm and G. Yang, Cutting through form factors and cross sections of non-protected operators in \( \mathcal{N}=4 \) SYM, JHEP 06 (2015) 156 [arXiv:1410.8485] [INSPIRE].
  42. [42]
    W. Siegel, Inconsistency of Supersymmetric Dimensional Regularization, Phys. Lett. B 94 (1980) 37 [INSPIRE].
  43. [43]
    L.V. Avdeev, G.A. Chochia and A.A. Vladimirov, On the Scope of Supersymmetric Dimensional Regularization, Phys. Lett. B 105 (1981) 272 [INSPIRE].
  44. [44]
    L.V. Avdeev, Noninvariance of Regularization by Dimensional Reduction: An Explicit Example of Supersymmetry Breaking, Phys. Lett. B 117 (1982) 317 [INSPIRE].
  45. [45]
    L.V. Avdeev and A.A. Vladimirov, Dimensional Regularization and Supersymmetry, Nucl. Phys. B 219 (1983) 262 [INSPIRE].
  46. [46]
    J.A. Minahan and K. Zarembo, The Bethe ansatz for \( \mathcal{N}=4 \) superYang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [INSPIRE].
  47. [47]
    B. Guo, Lollipop diagrams in defect \( \mathcal{N}=4 \) super Yang-Mills theory, MSc Thesis, University of British Columbia, Vancouver Canada (2017).Google Scholar
  48. [48]
    M. de Leeuw, A.C. Ipsen, C. Kristjansen and M. Wilhelm, One-loop Wilson loops and the particle-interface potential in AdS/dCFT, Phys. Lett. B 768 (2017) 192 [arXiv:1608.04754] [INSPIRE].
  49. [49]
    J. Aguilera-Damia, D.H. Correa and V.I. Giraldo-Rivera, Circular Wilson loops in defect Conformal Field Theory, JHEP 03 (2017) 023 [arXiv:1612.07991] [INSPIRE].
  50. [50]
    M. Preti, D. Trancanelli and E. Vescovi, Quark-antiquark potential in defect conformal field theory, JHEP 10 (2017) 079 [arXiv:1708.04884] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    E. Widen, Two-point functions of SU(2)-subsector and length-two operators in dCFT, Phys. Lett. B 773 (2017) 435 [arXiv:1705.08679] [INSPIRE].
  52. [52]
    L. Piroli, B. Pozsgay and E. Vernier, What is an integrable quench?, Nucl. Phys. B 925 (2017) 362 [arXiv:1709.04796] [INSPIRE].
  53. [53]
    M. de Leeuw, C. Kristjansen and K.E. Vardinghus, work in progress.Google Scholar
  54. [54]
    M. de Leeuw, C. Kristjansen and G. Linardopoulos, One-point functions of non-protected operators in the SO(5) symmetric D3-D7 dCFT, J. Phys. A 50 (2017) 254001 [arXiv:1612.06236] [INSPIRE].
  55. [55]
    B. de Wit, J. Hoppe and H. Nicolai, On the Quantum Mechanics of Supermembranes, Nucl. Phys. B 305 (1988) 545 [INSPIRE].
  56. [56]
    J. Hoppe, Quantum theory of a massless relativistic surface and a two-dimensional bound state problem, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge U.S.A. (1982),

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Aleix Gimenez Grau
    • 1
  • Charlotte Kristjansen
    • 1
  • Matthias Volk
    • 1
    Email author
  • Matthias Wilhelm
    • 1
  1. 1.Niels Bohr InstituteCopenhagen UniversityCopenhagen ØDenmark

Personalised recommendations