# Double-trace flows and the swampland

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## 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* CFT_{3} with an Einstein gravity dual.

## Keywords

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

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## References

- [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]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].ADSzbMATHGoogle Scholar - [3]E. Witten,
*Anti-de Sitter space and holography*,*Adv. Theor. Math. Phys.***2**(1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [4]I. Heemskerk, J. Penedones, J. Polchinski and J. Sully,
*Holography from conformal field theory*,*JHEP***10**(2009) 079 [arXiv:0907.0151] [INSPIRE].ADSMathSciNetGoogle Scholar - [5]
- [6]
- [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].ADSMathSciNetzbMATHGoogle Scholar - [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].ADSMathSciNetzbMATHGoogle Scholar - [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].ADSMathSciNetzbMATHGoogle Scholar - [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.ADSMathSciNetGoogle Scholar - [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].zbMATHGoogle Scholar - [12]M. Berkooz and S.-J. Rey,
*Nonsupersymmetric stable vacua of M-theory*,*JHEP***01**(1999) 014 [hep-th/9807200] [INSPIRE].ADSGoogle Scholar - [13]
- [14]A. Armoni and A. Naqvi,
*A non-supersymmetric large-N*3*D CFT and its gravity dual*,*JHEP***09**(2008) 119 [arXiv:0806.4068] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [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]D. Gaiotto and A. Tomasiello,
*The gauge dual of Romans mass*,*JHEP***01**(2010) 015 [arXiv:0901.0969] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [17]
- [18]O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena,
*N*= 6*superconformal Chern-Simons-matter theories, M*2*-branes and their gravity duals*,*JHEP***10**(2008) 091 [arXiv:0806.1218] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [19]S. Giombi, V. Kirilin and E. Perlmutter,
*Double-trace deformations of conformal correlations*, arXiv:1801.01477 [INSPIRE]. - [20]S. Caron-Huot,
*Analyticity in spin in conformal theories*,*JHEP***09**(2017) 078 [arXiv:1703.00278] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [21]I.R. Klebanov and E. Witten,
*AdS/CFT correspondence and symmetry breaking*,*Nucl. Phys.***B 556**(1999) 89 [hep-th/9905104] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [22]E. Witten,
*Multitrace operators, boundary conditions and AdS/CFT correspondence*, hep-th/0112258 [INSPIRE]. - [23]W. Mueck,
*An Improved correspondence formula for AdS/CFT with multitrace operators*,*Phys. Lett.***B 531**(2002) 301 [hep-th/0201100] [INSPIRE].ADSGoogle Scholar - [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].ADSGoogle Scholar - [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]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].ADSMathSciNetzbMATHGoogle Scholar - [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].ADSMathSciNetGoogle Scholar - [28]D.E. Diaz and H. Dorn,
*Partition functions and double-trace deformations in AdS/CFT*,*JHEP***05**(2007) 046 [hep-th/0702163] [INSPIRE].ADSMathSciNetGoogle Scholar - [29]T. Hartman, C.A. Keller and B. Stoica,
*Universal spectrum of*2*d conformal field theory in the large c limit*,*JHEP***09**(2014) 118 [arXiv:1405.5137] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [30]J.M. Maldacena, J. Michelson and A. Strominger,
*Anti-de Sitter fragmentation*,*JHEP***02**(1999) 011 [hep-th/9812073] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [31]N. Seiberg and E. Witten,
*The D*1*/D*5*system and singular CFT*,*JHEP***04**(1999) 017 [hep-th/9903224] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [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]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]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]P. Breitenlohner and D.Z. Freedman,
*Positive energy in Anti-de Sitter backgrounds and gauged extended supergravity*,*Phys. Lett.***B 115**(1982) 197.ADSMathSciNetGoogle Scholar - [36]C. Cordova, T.T. Dumitrescu and K. Intriligator,
*Multiplets of superconformal symmetry in diverse dimensions*, arXiv:1612.00809 [INSPIRE]. - [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].ADSzbMATHGoogle Scholar - [38]M.J. Strassler,
*Nonsupersymmetric theories with light scalar fields and large hierarchies*, hep-th/0309122 [INSPIRE]. - [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].ADSMathSciNetzbMATHGoogle Scholar - [40]I.R. Klebanov and G. Torri,
*M*2*-branes and AdS/CFT*,*Int. J. Mod. Phys.***A 25**(2010) 332 [arXiv:0909.1580] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [41]J. Bagger, N. Lambert, S. Mukhi and C. Papageorgakis,
*Multiple membranes in M-theory*,*Phys. Rept.***527**(2013) 1 [arXiv:1203.3546] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [42]D.Z. Freedman and S.S. Pufu,
*The holography of F -maximization*,*JHEP***03**(2014) 135 [arXiv:1302.7310] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [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].ADSMathSciNetzbMATHGoogle Scholar - [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].ADSGoogle Scholar - [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].ADSMathSciNetGoogle Scholar - [46]M.J. Duff, B.E.W. Nilsson and C.N. Pope,
*Kaluza-Klein supergravity*,*Phys. Rept.***130**(1986) 1 [INSPIRE].ADSMathSciNetGoogle Scholar - [47]J.T. Liu and W. Zhao,
*One-loop supergravity on*AdS_{4}×*S*^{7}/ℤ_{k}*and comparison with ABJM theory*,*JHEP***11**(2016) 099 [arXiv:1609.02558] [INSPIRE].ADSGoogle Scholar - [48]G. Gur-Ari and R. Yacoby,
*Three dimensional bosonization from supersymmetry*,*JHEP***11**(2015) 013 [arXiv:1507.04378] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [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].ADSMathSciNetGoogle Scholar - [50]A. Dymarsky, I.R. Klebanov and R. Roiban,
*Perturbative gauge theory and closed string tachyons*,*JHEP***11**(2005) 038 [hep-th/0509132] [INSPIRE].ADSMathSciNetGoogle Scholar - [51]E. Pomoni and L. Rastelli,
*Large-N field theory and AdS tachyons*,*JHEP***04**(2009) 020 [arXiv:0805.2261] [INSPIRE].ADSMathSciNetGoogle Scholar - [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].MathSciNetzbMATHGoogle Scholar - [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].ADSzbMATHGoogle Scholar - [54]O. Aharony, O. Bergman and D.L. Jafferis,
*Fractional M*2*-branes*,*JHEP***11**(2008) 043 [arXiv:0807.4924] [INSPIRE].ADSMathSciNetGoogle Scholar - [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].ADSzbMATHGoogle Scholar - [56]D. Fabbri et al., 3
*D superconformal theories from Sasakian seven manifolds: new nontrivial evidences for AdS*_{4}*/CFT*_{3},*Nucl. Phys.***B 577**(2000) 547 [hep-th/9907219] [INSPIRE].ADSzbMATHGoogle Scholar - [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]X. Dong, D.Z. Freedman and Y. Zhao,
*Explicitly broken supersymmetry with exactly massless moduli*,*JHEP***06**(2016) 090 [arXiv:1410.2257] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [59]X. Dong, D.Z. Freedman and Y. Zhao,
*AdS/CFT and the little hierarchy problem*, arXiv:1510.01741 [INSPIRE]. - [60]E. Witten, SL(2, ℤ)
*action on three-dimensional conformal field theories with Abelian symmetry*, hep-th/0307041 [INSPIRE]. - [61]A.A. Tseytlin and K. Zarembo,
*Effective potential in nonsupersymmetric*SU(*N*) × SU(*N*)*gauge theory and interactions of type 0 D*3*-branes*,*Phys. Lett.***B 457**(1999) 77 [hep-th/9902095] [INSPIRE].ADSzbMATHGoogle Scholar - [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].ADSGoogle Scholar - [63]A. Bernamonti and B. Craps,
*D-brane potentials from multi-trace deformations in AdS/CFT*,*JHEP***08**(2009) 112 [arXiv:0907.0889] [INSPIRE].ADSMathSciNetGoogle Scholar - [64]I.R. Klebanov, S.S. Pufu and B.R. Safdi,
*F-theorem without supersymmetry*,*JHEP***10**(2011) 038 [arXiv:1105.4598] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [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].ADSMathSciNetzbMATHGoogle Scholar - [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]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].zbMATHGoogle Scholar - [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].ADSMathSciNetzbMATHGoogle Scholar - [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].ADSzbMATHGoogle Scholar - [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].ADSMathSciNetzbMATHGoogle Scholar - [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].ADSMathSciNetzbMATHGoogle Scholar - [72]A.C. Petkou,
*Evaluating the AdS dual of the critical O*(*N*)*vector model*,*JHEP***03**(2003) 049 [hep-th/0302063] [INSPIRE].ADSMathSciNetGoogle Scholar - [73]E. Sezgin and P. Sundell,
*Holography in*4*D (super) higher spin theories and a test via cubic scalar couplings*,*JHEP***07**(2005) 044 [hep-th/0305040] [INSPIRE].ADSMathSciNetGoogle Scholar - [74]N. Yamatsu,
*Finite-dimensional Lie algebras and their representations for unified model building*, arXiv:1511.08771 [INSPIRE]. - [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].ADSMathSciNetzbMATHGoogle Scholar - [76]M. Nirschl and H. Osborn,
*Superconformal Ward identities and their solution*,*Nucl. Phys.***B 711**(2005) 409 [hep-th/0407060] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [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].ADSMathSciNetGoogle Scholar - [78]F. Aprile, J.M. Drummond, P. Heslop and H. Paul,
*Unmixing supergravity*,*JHEP***02**(2018) 133 [arXiv:1706.08456] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [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].ADSGoogle Scholar - [80]D. Li, D. Meltzer and D. Poland,
*Non-abelian binding energies from the lightcone bootstrap*,*JHEP***02**(2016) 149 [arXiv:1510.07044] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [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].ADSMathSciNetzbMATHGoogle Scholar - [82]Z. Komargodski and A. Zhiboedov,
*Convexity and liberation at large spin*,*JHEP***11**(2013) 140 [arXiv:1212.4103] [INSPIRE].ADSGoogle Scholar - [83]L.F. Alday and A. Zhiboedov,
*An algebraic approach to the analytic bootstrap*,*JHEP***04**(2017) 157 [arXiv:1510.08091] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [84]D. Li, D. Meltzer and D. Poland,
*Conformal collider physics from the lightcone bootstrap*,*JHEP***02**(2016) 143 [arXiv:1511.08025] [INSPIRE].ADSGoogle Scholar - [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].ADSMathSciNetzbMATHGoogle Scholar - [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].ADSMathSciNetzbMATHGoogle Scholar - [87]A.L. Fitzpatrick and D. Shih,
*Anomalous dimensions of non-chiral operators from AdS/CFT*,*JHEP***10**(2011) 113 [arXiv:1104.5013] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [88]S. Giombi, C. Sleight and M. Taronna,
*Spinning AdS loop diagrams: two point functions*, arXiv:1708.08404 [INSPIRE]. - [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]S. Giombi et al.,
*Chern-Simons theory with vector fermion matter*,*Eur. Phys. J.***C 72**(2012) 2112 [arXiv:1110.4386] [INSPIRE].ADSGoogle Scholar - [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].ADSMathSciNetzbMATHGoogle Scholar - [92]A. Karch and D. Tong,
*Particle-vortex duality from*3*d bosonization*,*Phys. Rev.***X 6**(2016) 031043 [arXiv:1606.01893] [INSPIRE].Google Scholar - [93]J. Murugan and H. Nastase,
*Particle-vortex duality in topological insulators and superconductors*,*JHEP***05**(2017) 159 [arXiv:1606.01912] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [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].ADSMathSciNetzbMATHGoogle Scholar - [95]J.M. Maldacena, A. Strominger and E. Witten,
*Black hole entropy in M-theory*,*JHEP***12**(1997) 002 [hep-th/9711053] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [96]F. Larsen,
*The Perturbation spectrum of black holes in N*= 8*supergravity*,*Nucl. Phys.***B 536**(1998) 258 [hep-th/9805208] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [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].ADSzbMATHGoogle Scholar - [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].MathSciNetzbMATHGoogle Scholar - [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]A.L. Fitzpatrick and J. Kaplan,
*Unitarity and the holographic S-matrix*,*JHEP***10**(2012) 032 [arXiv:1112.4845] [INSPIRE].ADSzbMATHGoogle Scholar - [101]S. Kachru and E. Silverstein, 4
*D conformal theories and strings on orbifolds*,*Phys. Rev. Lett.***80**(1998) 4855 [hep-th/9802183] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [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].ADSMathSciNetzbMATHGoogle Scholar - [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