Advertisement

Double-trace flows and the swampland

  • Simone Giombi
  • Eric Perlmutter
Open Access
Regular Article - Theoretical Physics
  • 40 Downloads

Abstract

We explore the idea that large N, non-supersymmetric conformal field theories with a parametrically large gap to higher spin single-trace operators may be obtained as infrared fixed points of relevant double-trace deformations of superconformal field theories. After recalling the AdS interpretation and some potential pathologies of such flows, we introduce a concrete example that appears to avoid them: the ABJM theory at finite k, deformed by \( {\displaystyle \int {\mathcal{O}}^2} \), where \( \mathcal{O} \) is the superconformal primary in the stress-tensor multiplet. We address its relation to recent conjectures based on weak gravity bounds, and discuss the prospects for a wider class of similarly viable flows. Next, we proceed to analyze the spectrum and correlation functions of the putative IR CFT, to leading non-trivial order in 1/N. This includes analytic computations of the change under double-trace flow of connected four-point functions of ABJM superconformal primaries; and of the IR anomalous dimensions of infinite classes of double-trace composite operators. These would be the first analytic results for anomalous dimensions of finite-spin composite operators in any large N CFT3 with an Einstein gravity dual.

Keywords

1/N Expansion AdS-CFT Correspondence Conformal Field Theory Field Theories in Higher Dimensions 

Notes

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.

References

  1. [1]
    J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [INSPIRE].
  2. [2]
    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  3. [3]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from conformal field theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    H. Ooguri and C. Vafa, Non-supersymmetric AdS and the swampland, arXiv:1610.01533 [INSPIRE].
  6. [6]
    B. Freivogel and M. Kleban, Vacua morghulis, arXiv:1610.04564 [INSPIRE].
  7. [7]
    X.O. Camanho, J.D. Edelstein, J. Maldacena and A. Zhiboedov, Causality constraints on corrections to the graviton three-point coupling, JHEP 02 (2016) 020 [arXiv:1407.5597] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    N. Afkhami-Jeddi, T. Hartman, S. Kundu and A. Tajdini, Einstein gravity 3-point functions from conformal field theory, JHEP 12 (2017) 049 [arXiv:1610.09378] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    M.S. Costa, T. Hansen and J. Penedones, Bounds for OPE coefficients on the Regge trajectory, JHEP 10 (2017) 197 [arXiv:1707.07689] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    M.J. Duff, B.E.W. Nilsson, and C. N. Pope, The criterion for vacuum stability in Kaluza-Klein supergravity, Phys. Lett. B 139 (1984) 154.ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    J. Distler and F. Zamora, Nonsupersymmetric conformal field theories from stable Anti-de Sitter spaces, Adv. Theor. Math. Phys. 2 (1999) 1405 [hep-th/9810206] [INSPIRE].CrossRefzbMATHGoogle Scholar
  12. [12]
    M. Berkooz and S.-J. Rey, Nonsupersymmetric stable vacua of M-theory, JHEP 01 (1999) 014 [hep-th/9807200] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    A. Murugan, Renormalization group flows in gauge-gravity duality, arXiv:1610.03166 [INSPIRE].
  14. [14]
    A. Armoni and A. Naqvi, A non-supersymmetric large-N 3D CFT and its gravity dual, JHEP 09 (2008) 119 [arXiv:0806.4068] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    T. Fischbacher, K. Pilch and N.P. Warner, New supersymmetric and stable, non-supersymmetric phases in supergravity and holographic field theory, arXiv:1010.4910 [INSPIRE].
  16. [16]
    D. Gaiotto and A. Tomasiello, The gauge dual of Romans mass, JHEP 01 (2010) 015 [arXiv:0901.0969] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    T. Banks, Note on a paper by Ooguri and Vafa, arXiv:1611.08953 [INSPIRE].
  18. [18]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    S. Giombi, V. Kirilin and E. Perlmutter, Double-trace deformations of conformal correlations, arXiv:1801.01477 [INSPIRE].
  20. [20]
    S. Caron-Huot, Analyticity in spin in conformal theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    I.R. Klebanov and E. Witten, AdS/CFT correspondence and symmetry breaking, Nucl. Phys. B 556 (1999) 89 [hep-th/9905104] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    E. Witten, Multitrace operators, boundary conditions and AdS/CFT correspondence, hep-th/0112258 [INSPIRE].
  23. [23]
    W. Mueck, An Improved correspondence formula for AdS/CFT with multitrace operators, Phys. Lett. B 531 (2002) 301 [hep-th/0201100] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    M. Berkooz, A. Sever and A. Shomer, ‘Double trace’ deformations, boundary conditions and space-time singularities, JHEP 05 (2002) 034 [hep-th/0112264] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    S.S. Gubser and I. Mitra, Double trace operators and one loop vacuum energy in AdS/CFT, Phys. Rev. D 67 (2003) 064018 [hep-th/0210093] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  26. [26]
    S.S. Gubser and I.R. Klebanov, A universal result on central charges in the presence of double trace deformations, Nucl. Phys. B 656 (2003) 23 [hep-th/0212138] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    T. Hartman and L. Rastelli, Double-trace deformations, mixed boundary conditions and functional determinants in AdS/CFT, JHEP 01 (2008) 019 [hep-th/0602106] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    D.E. Diaz and H. Dorn, Partition functions and double-trace deformations in AdS/CFT, JHEP 05 (2007) 046 [hep-th/0702163] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    T. Hartman, C.A. Keller and B. Stoica, Universal spectrum of 2d conformal field theory in the large c limit, JHEP 09 (2014) 118 [arXiv:1405.5137] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    J.M. Maldacena, J. Michelson and A. Strominger, Anti-de Sitter fragmentation, JHEP 02 (1999) 011 [hep-th/9812073] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    N. Seiberg and E. Witten, The D1/D5 system and singular CFT, JHEP 04 (1999) 017 [hep-th/9903224] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    Y. Nakayama and Y. Nomura, Weak gravity conjecture in the AdS/CFT correspondence, Phys. Rev. D 92 (2015) 126006 [arXiv:1509.01647] [INSPIRE].ADSMathSciNetGoogle Scholar
  33. [33]
    A. Adams and E. Silverstein, Closed string tachyons, AdS/CFT and large-N QCD, Phys. Rev. D 64 (2001) 086001 [hep-th/0103220] [INSPIRE].ADSMathSciNetGoogle Scholar
  34. [34]
    G.T. Horowitz, J. Orgera and J. Polchinski, Nonperturbative Instability of AdS 5 × S 5 /Z k, Phys. Rev. D 77 (2008) 024004 [arXiv:0709.4262] [INSPIRE].ADSGoogle Scholar
  35. [35]
    P. Breitenlohner and D.Z. Freedman, Positive energy in Anti-de Sitter backgrounds and gauged extended supergravity, Phys. Lett. B 115 (1982) 197.ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    C. Cordova, T.T. Dumitrescu and K. Intriligator, Multiplets of superconformal symmetry in diverse dimensions, arXiv:1612.00809 [INSPIRE].
  37. [37]
    I.R. Klebanov and E. Witten, Superconformal field theory on three-branes at a Calabi-Yau singularity, Nucl. Phys. B 536 (1998) 199 [hep-th/9807080] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  38. [38]
    M.J. Strassler, Nonsupersymmetric theories with light scalar fields and large hierarchies, hep-th/0309122 [INSPIRE].
  39. [39]
    S.S. Gubser and I.R. Klebanov, A Universal result on central charges in the presence of double trace deformations, Nucl. Phys. B 656 (2003) 23 [hep-th/0212138] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    I.R. Klebanov and G. Torri, M2-branes and AdS/CFT, Int. J. Mod. Phys. A 25 (2010) 332 [arXiv:0909.1580] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    J. Bagger, N. Lambert, S. Mukhi and C. Papageorgakis, Multiple membranes in M-theory, Phys. Rept. 527 (2013) 1 [arXiv:1203.3546] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    D.Z. Freedman and S.S. Pufu, The holography of F -maximization, JHEP 03 (2014) 135 [arXiv:1302.7310] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    H. Osborn and A.C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    S.M. Chester, J. Lee, S.S. Pufu and R. Yacoby, The \( \mathcal{N}=8 \) superconformal bootstrap in three dimensions, JHEP 09 (2014) 143 [arXiv:1406.4814] [INSPIRE].ADSCrossRefGoogle Scholar
  45. [45]
    L. Castellani, R. D’Auria, P. Fré, K. Pilch and P. van Nieuwenhuizen, The bosonic mass formula for Freund-Rubin solutions of d = 11 supergravity on general coset manifolds, Class. Quant. Grav. 1 (1984) 339 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  46. [46]
    M.J. Duff, B.E.W. Nilsson and C.N. Pope, Kaluza-Klein supergravity, Phys. Rept. 130 (1986) 1 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  47. [47]
    J.T. Liu and W. Zhao, One-loop supergravity on AdS4 × S 7/ℤk and comparison with ABJM theory, JHEP 11 (2016) 099 [arXiv:1609.02558] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    G. Gur-Ari and R. Yacoby, Three dimensional bosonization from supersymmetry, JHEP 11 (2015) 013 [arXiv:1507.04378] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    A. Dymarsky, I.R. Klebanov and R. Roiban, Perturbative search for fixed lines in large-N gauge theories, JHEP 08 (2005) 011 [hep-th/0505099] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  50. [50]
    A. Dymarsky, I.R. Klebanov and R. Roiban, Perturbative gauge theory and closed string tachyons, JHEP 11 (2005) 038 [hep-th/0509132] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  51. [51]
    E. Pomoni and L. Rastelli, Large-N field theory and AdS tachyons, JHEP 04 (2009) 020 [arXiv:0805.2261] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  52. [52]
    D.Z. Freedman, S.S. Gubser, K. Pilch and N.P. Warner, Renormalization group flows from holography supersymmetry and a c theorem, Adv. Theor. Math. Phys. 3 (1999) 363 [hep-th/9904017] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    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].ADSCrossRefzbMATHGoogle Scholar
  54. [54]
    O. Aharony, O. Bergman and D.L. Jafferis, Fractional M2-branes, JHEP 11 (2008) 043 [arXiv:0807.4924] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  55. [55]
    D. Fabbri, P. Fré, L. Gualtieri and P. Termonia, M theory on AdS 4 × M 111 : the complete Osp(2|4) × SU(3) × SU(2) spectrum from harmonic analysis, Nucl. Phys. B 560 (1999) 617 [hep-th/9903036] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  56. [56]
    D. Fabbri et al., 3D superconformal theories from Sasakian seven manifolds: new nontrivial evidences for AdS 4 /CFT 3, Nucl. Phys. B 577 (2000) 547 [hep-th/9907219] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  57. [57]
    D. Martelli and J. Sparks, Moduli spaces of Chern-Simons quiver gauge theories and AdS 4 /CFT 3, Phys. Rev. D 78 (2008) 126005 [arXiv:0808.0912] [INSPIRE].ADSGoogle Scholar
  58. [58]
    X. Dong, D.Z. Freedman and Y. Zhao, Explicitly broken supersymmetry with exactly massless moduli, JHEP 06 (2016) 090 [arXiv:1410.2257] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    X. Dong, D.Z. Freedman and Y. Zhao, AdS/CFT and the little hierarchy problem, arXiv:1510.01741 [INSPIRE].
  60. [60]
    E. Witten, SL(2, ℤ) action on three-dimensional conformal field theories with Abelian symmetry, hep-th/0307041 [INSPIRE].
  61. [61]
    A.A. Tseytlin and K. Zarembo, Effective potential in nonsupersymmetric SU(N) × SU(N) gauge theory and interactions of type 0 D3-branes, Phys. Lett. B 457 (1999) 77 [hep-th/9902095] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  62. [62]
    A. Adams, J. Polchinski and E. Silverstein, Don’t panic! Closed string tachyons in ALE space-times, JHEP 10 (2001) 029 [hep-th/0108075] [INSPIRE].ADSCrossRefGoogle Scholar
  63. [63]
    A. Bernamonti and B. Craps, D-brane potentials from multi-trace deformations in AdS/CFT, JHEP 08 (2009) 112 [arXiv:0907.0889] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  64. [64]
    I.R. Klebanov, S.S. Pufu and B.R. Safdi, F-theorem without supersymmetry, JHEP 10 (2011) 038 [arXiv:1105.4598] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  65. [65]
    N. Drukker, M. Mariño and P. Putrov, From weak to strong coupling in ABJM theory, Commun. Math. Phys. 306 (2011) 511 [arXiv:1007.3837] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  66. [66]
    E. D’Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Extremal correlators in the AdS/CFT correspondence, hep-th/9908160 [INSPIRE].
  67. [67]
    E. D’Hoker and B. Pioline, Near extremal correlators and generalized consistent truncation for AdS(4|7) × S 7|4, JHEP 07 (2000) 021 [hep-th/0006103] [INSPIRE].CrossRefzbMATHGoogle Scholar
  68. [68]
    F. Bastianelli and R. Zucchini, Three point functions of chiral primary operators in D = 3, N = 8 and D = 6, N = (2, 0) SCFT at large-N, Phys. Lett. B 467 (1999) 61 [hep-th/9907047] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  69. [69]
    D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Correlation functions in the CFT d /AdS d+1 correspondence, Nucl. Phys. B 546 (1999) 96 [hep-th/9804058] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  70. [70]
    D.Z. Freedman, K. Pilch, S.S. Pufu and N.P. Warner, Boundary terms and three-point functions: an AdS/CFT puzzle resolved, JHEP 06 (2017) 053 [arXiv:1611.01888] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  71. [71]
    A. Petkou, Conserved currents, consistency relations and operator product expansions in the conformally invariant O(N) vector model, Annals Phys. 249 (1996) 180 [hep-th/9410093] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  72. [72]
    A.C. Petkou, Evaluating the AdS dual of the critical O(N) vector model, JHEP 03 (2003) 049 [hep-th/0302063] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  73. [73]
    E. Sezgin and P. Sundell, Holography in 4D (super) higher spin theories and a test via cubic scalar couplings, JHEP 07 (2005) 044 [hep-th/0305040] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  74. [74]
    N. Yamatsu, Finite-dimensional Lie algebras and their representations for unified model building, arXiv:1511.08771 [INSPIRE].
  75. [75]
    S. Ferrara and E. Sokatchev, Universal properties of superconformal OPEs for 1/2 BPS operators in 3 ≤ D ≤ 6, New J. Phys. 4 (2002) 2 [hep-th/0110174] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  76. [76]
    M. Nirschl and H. Osborn, Superconformal Ward identities and their solution, Nucl. Phys. B 711 (2005) 409 [hep-th/0407060] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  77. [77]
    F.A. Dolan, L. Gallot and E. Sokatchev, On four-point functions of 1/2-BPS operators in general dimensions, JHEP 09 (2004) 056 [hep-th/0405180] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  78. [78]
    F. Aprile, J.M. Drummond, P. Heslop and H. Paul, Unmixing supergravity, JHEP 02 (2018) 133 [arXiv:1706.08456] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  79. [79]
    L.F. Alday and A. Bissi, Loop corrections to supergravity on AdS 5 × S 5, Phys. Rev. Lett. 119 (2017) 171601 [arXiv:1706.02388] [INSPIRE].ADSCrossRefGoogle Scholar
  80. [80]
    D. Li, D. Meltzer and D. Poland, Non-abelian binding energies from the lightcone bootstrap, JHEP 02 (2016) 149 [arXiv:1510.07044] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  81. [81]
    A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The analytic bootstrap and AdS superhorizon locality, JHEP 12 (2013) 004 [arXiv:1212.3616] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  82. [82]
    Z. Komargodski and A. Zhiboedov, Convexity and liberation at large spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].ADSCrossRefGoogle Scholar
  83. [83]
    L.F. Alday and A. Zhiboedov, An algebraic approach to the analytic bootstrap, JHEP 04 (2017) 157 [arXiv:1510.08091] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  84. [84]
    D. Li, D. Meltzer and D. Poland, Conformal collider physics from the lightcone bootstrap, JHEP 02 (2016) 143 [arXiv:1511.08025] [INSPIRE].ADSCrossRefGoogle Scholar
  85. [85]
    L. Cornalba, M.S. Costa, J. Penedones and R. Schiappa, Eikonal approximation in AdS/CFT: conformal partial waves and finite N four-point functions, Nucl. Phys. B 767 (2007) 327 [hep-th/0611123] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  86. [86]
    A.L. Fitzpatrick, E. Katz, D. Poland and D. Simmons-Duffin, Effective Conformal Theory and the Flat-Space Limit of AdS, JHEP 07 (2011) 023 [arXiv:1007.2412] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  87. [87]
    A.L. Fitzpatrick and D. Shih, Anomalous dimensions of non-chiral operators from AdS/CFT, JHEP 10 (2011) 113 [arXiv:1104.5013] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  88. [88]
    S. Giombi, C. Sleight and M. Taronna, Spinning AdS loop diagrams: two point functions, arXiv:1708.08404 [INSPIRE].
  89. [89]
    S. Giombi and X. Yin, On higher spin gauge theory and the critical O(N) model, Phys. Rev. D 85 (2012) 086005 [arXiv:1105.4011] [INSPIRE].ADSGoogle Scholar
  90. [90]
    S. Giombi et al., Chern-Simons theory with vector fermion matter, Eur. Phys. J. C 72 (2012) 2112 [arXiv:1110.4386] [INSPIRE].ADSCrossRefGoogle Scholar
  91. [91]
    O. Aharony, G. Gur-Ari and R. Yacoby, D = 3 bosonic vector models coupled to Chern-Simons gauge theories, JHEP 03 (2012) 037 [arXiv:1110.4382] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  92. [92]
    A. Karch and D. Tong, Particle-vortex duality from 3d bosonization, Phys. Rev. X 6 (2016) 031043 [arXiv:1606.01893] [INSPIRE].CrossRefGoogle Scholar
  93. [93]
    J. Murugan and H. Nastase, Particle-vortex duality in topological insulators and superconductors, JHEP 05 (2017) 159 [arXiv:1606.01912] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  94. [94]
    N. Seiberg, T. Senthil, C. Wang and E. Witten, A duality web in 2 + 1 dimensions and condensed matter physics, Annals Phys. 374 (2016) 395 [arXiv:1606.01989] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  95. [95]
    J.M. Maldacena, A. Strominger and E. Witten, Black hole entropy in M-theory, JHEP 12 (1997) 002 [hep-th/9711053] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  96. [96]
    F. Larsen, The Perturbation spectrum of black holes in N = 8 supergravity, Nucl. Phys. B 536 (1998) 258 [hep-th/9805208] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  97. [97]
    A. Fujii, R. Kemmoku and S. Mizoguchi, D = 5 simple supergravity on AdS 3 × S 2 and N = 4 superconformal field theory, Nucl. Phys. B 574 (2000) 691 [hep-th/9811147] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  98. [98]
    J. de Boer, S. El-Showk, I. Messamah and D. Van den Bleeken, A bound on the entropy of supergravity?, JHEP 02 (2010) 062 [arXiv:0906.0011] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  99. [99]
    K. Diab et al., On C J and C T in the Gross-Neveu and O(N) models, J. Phys. A 49 (2016) 405402 [arXiv:1601.07198] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  100. [100]
    A.L. Fitzpatrick and J. Kaplan, Unitarity and the holographic S-matrix, JHEP 10 (2012) 032 [arXiv:1112.4845] [INSPIRE].ADSCrossRefGoogle Scholar
  101. [101]
    S. Kachru and E. Silverstein, 4D conformal theories and strings on orbifolds, Phys. Rev. Lett. 80 (1998) 4855 [hep-th/9802183] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  102. [102]
    A.E. Lawrence, N. Nekrasov and C. Vafa, On conformal field theories in four-dimensions, Nucl. Phys. B 533 (1998) 199 [hep-th/9803015] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  103. [103]
    P. Liendo, Orientifold daughter of N = 4 SYM and double-trace running, Phys. Rev. D 86 (2012) 105032 [arXiv:1107.3125] [INSPIRE].ADSGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Department of PhysicsPrinceton UniversityPrincetonU.S.A.
  2. 2.Walter Burke Institute for Theoretical PhysicsCalifornia Institute of TechnologyPasadenaU.S.A.

Personalised recommendations