# Universal structure of covariant holographic two-point functions in massless higher-order gravities

## Abstract

We consider massless higher-order gravities in general *D* = *d* + 1 dimensions, which are Einstein gravity extended with higher-order curvature invariants in such a way that the linearized spectrum around the AdS vacua involves only the massless graviton. We derive the covariant holographic two-point functions and find that they have a universal structure. In particular, the theory-dependent overall coefficient factor \( {\mathcal{C}}_T \) can be universally expressed by \( \left(d - 1\right){\mathcal{C}}_T = \ell \left(\partial a/\partial \ell \right) \), where *a* is the holographic *a*-charge and *ℓ* is the AdS radius. We verify this relation in quasi-topological Ricci polynomial, Einstein-Gauss-Bonnet, Einstein-Lovelock and Einstein cubic gravities. In *d* = 4, we also find an intriguing relation between the holographic *c* and *a* charges, namely \( c=\frac{1}{3}\ell \left(\partial a/\partial \ell \right) \), which also implies \( {\mathcal{C}}_T=c \).

## Keywords

AdS-CFT Correspondence Classical Theories of Gravity## Notes

### **Open Access**

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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 [hep-th/9711200] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar - [2]J. de Boer,
*The holographic renormalization group*,*Fortsch. Phys.***49**(2001) 339 [hep-th/0101026] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [3]M. Bianchi, D.Z. Freedman and K. Skenderis,
*Holographic renormalization*,*Nucl. Phys.***B 631**(2002) 159 [hep-th/0112119] [INSPIRE]. - [4]K. Skenderis,
*Lecture notes on holographic renormalization*,*Class. Quant. Grav.***19**(2002) 5849 [hep-th/0209067] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar - [5]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]. - [6]E. Witten,
*Anti-de Sitter space and holography*,*Adv. Theor. Math. Phys.***2**(1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [7]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]. - [8]W. Mueck and K.S. Viswanathan,
*Conformal field theory correlators from classical scalar field theory on AdS*_{d+1},*Phys. Rev.***D 58**(1998) 041901 [hep-th/9804035] [INSPIRE]. - [9]H. Liu and A.A. Tseytlin,
*D*= 4*super Yang-Mills, D*= 5*gauged supergravity and D*= 4*conformal supergravity*,*Nucl. Phys.***B 533**(1998) 88 [hep-th/9804083] [INSPIRE]. - [10]E. Keski-Vakkuri,
*Bulk and boundary dynamics in BTZ black holes*,*Phys. Rev.***D 59**(1999) 104001 [hep-th/9808037] [INSPIRE]. - [11]K.S. Stelle,
*Renormalization of higher derivative quantum gravity*,*Phys. Rev.***D 16**(1977) 953 [INSPIRE]. - [12]H. Lü and C.N. Pope,
*Critical gravity in four dimensions*,*Phys. Rev. Lett.***106**(2011) 181302 [arXiv:1101.1971] [INSPIRE].ADSCrossRefGoogle Scholar - [13]S. Deser, H. Liu, H. Lü, C.N. Pope, T.C. Sisman and B. Tekin,
*Critical points of D-dimensional extended gravities*,*Phys. Rev.***D 83**(2011) 061502 [arXiv:1101.4009] [INSPIRE]. - [14]N. Johansson, A. Naseh and T. Zojer,
*Holographic two-point functions for*4*d log-gravity*,*JHEP***09**(2012) 114 [arXiv:1205.5804] [INSPIRE]. - [15]A. Ghodsi, B. Khavari and A. Naseh,
*Holographic two-point functions in conformal gravity*,*JHEP***01**(2015) 137 [arXiv:1411.3158] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [16]D. Lovelock,
*The Einstein tensor and its generalizations*,*J. Math. Phys.***12**(1971) 498 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [17]B. Tekin,
*Particle content of quadratic and f*(*R*_{μνσρ})*theories in (A)dS*,*Phys. Rev.***D 93**(2016) 101502 [arXiv:1604.00891] [INSPIRE]. - [18]P. Bueno and P.A. Cano,
*Einsteinian cubic gravity*,*Phys. Rev.***D 94**(2016) 104005 [arXiv:1607.06463] [INSPIRE]. - [19]P. Bueno, P.A. Cano, V.S. Min and M.R. Visser,
*Aspects of general higher-order gravities*,*Phys. Rev.***D 95**(2017) 044010 [arXiv:1610.08519] [INSPIRE]. - [20]Y.-Z. Li, H.-S. Liu and H. Lü,
*Quasi-topological Ricci polynomial gravities*,*JHEP***02**(2018) 166 [arXiv:1708.07198] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar - [21]M.H. Dehghani, A. Bazrafshan, R.B. Mann, M.R. Mehdizadeh, M. Ghanaatian and M.H. Vahidinia,
*Black holes in quartic quasitopological gravity*,*Phys. Rev.***D 85**(2012) 104009 [arXiv:1109.4708] [INSPIRE]. - [22]A. Karasu, E. Kenar and B. Tekin,
*Minimal extension of Einstein’s theory: the quartic gravity*,*Phys. Rev.***D 93**(2016) 084040 [arXiv:1602.02567] [INSPIRE]. - [23]P. Bueno, P.A. Cano, A.O. Lasso and P.F. Ramírez,
*f(Lovelock) theories of gravity*,*JHEP***04**(2016) 028 [arXiv:1602.07310] [INSPIRE]. - [24]A. Cisterna, L. Guajardo, M. Hassaine and J. Oliva,
*Quintic quasi-topological gravity*,*JHEP***04**(2017) 066 [arXiv:1702.04676] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [25]R.A. Hennigar, D. Kubizňák and R.B. Mann,
*Generalized quasitopological gravity*,*Phys. Rev.***D 95**(2017) 104042 [arXiv:1703.01631] [INSPIRE]. - [26]J. Ahmed, R.A. Hennigar, R.B. Mann and M. Mir,
*Quintessential quartic quasi-topological quartet*,*JHEP***05**(2017) 134 [arXiv:1703.11007] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [27]X.-H. Feng, H. Huang, S.-L. Li, H. Lü and H. Wei,
*Cosmological time crystals from Einstein-cubic gravities*, arXiv:1807.01720 [INSPIRE]. - [28]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 - [29]J. Erdmenger and H. Osborn,
*Conserved currents and the energy momentum tensor in conformally invariant theories for general dimensions*,*Nucl. Phys.***B 483**(1997) 431 [hep-th/9605009] [INSPIRE]. - [30]C. Corianò, L. Delle Rose, E. Mottola and M. Serino,
*Graviton vertices and the mapping of anomalous correlators to momentum space for a general conformal field theory*,*JHEP***08**(2012) 147 [arXiv:1203.1339] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [31]A. Buchel, J. Escobedo, R.C. Myers, M.F. Paulos, A. Sinha and M. Smolkin,
*Holographic GB gravity in arbitrary dimensions*,*JHEP***03**(2010) 111 [arXiv:0911.4257] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [32]Y.-Z. Li, H. Lü and J.-B. Wu,
*Causality and a-theorem constraints on Ricci polynomial and Riemann cubic gravities*,*Phys. Rev.***D 97**(2018) 024023 [arXiv:1711.03650] [INSPIRE]. - [33]M. Henningson and K. Skenderis,
*The holographic Weyl anomaly*,*JHEP***07**(1998) 023 [hep-th/9806087] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [34]M. Henningson and K. Skenderis,
*Holography and the Weyl anomaly*,*Fortsch. Phys.***48**(2000) 125 [hep-th/9812032] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [35]C. Imbimbo, A. Schwimmer, S. Theisen and S. Yankielowicz,
*Diffeomorphisms and holographic anomalies*,*Class. Quant. Grav.***17**(2000) 1129 [hep-th/9910267] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [36]S. Nojiri and S.D. Odintsov,
*On the conformal anomaly from higher derivative gravity in AdS/CFT correspondence*,*Int. J. Mod. Phys.***A 15**(2000) 413 [hep-th/9903033] [INSPIRE]. - [37]M. Blau, K.S. Narain and E. Gava,
*On subleading contributions to the AdS/CFT trace anomaly*,*JHEP***09**(1999) 018 [hep-th/9904179] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [38]R.C. Myers and A. Sinha,
*Seeing a c-theorem with holography*,*Phys. Rev.***D 82**(2010) 046006 [arXiv:1006.1263] [INSPIRE]. - [39]R.C. Myers and A. Sinha,
*Holographic c-theorems in arbitrary dimensions*,*JHEP***01**(2011) 125 [arXiv:1011.5819] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [40]S. de Haro, S.N. Solodukhin and K. Skenderis,
*Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence*,*Commun. Math. Phys.***217**(2001) 595 [hep-th/0002230] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [41]D. Marolf, W. Kelly and S. Fischetti,
*Conserved charges in asymptotically (locally) AdS spacetimes*, in*Springer handbook of spacetime*, Springer, Berlin, Heidelberg, Germany, (2014), pg. 381 [arXiv:1211.6347] [INSPIRE]. - [42]Y.-X. Chen, H. Lü and K.-N. Shao,
*Linearized modes in extended and critical gravities*,*Class. Quant. Grav.***29**(2012) 085017 [arXiv:1108.5184] [INSPIRE]. - [43]H. Lü and K.-N. Shao,
*Solutions of free higher spins in AdS*,*Phys. Lett.***B 706**(2011) 106 [arXiv:1110.1138] [INSPIRE]. - [44]R.C. Myers and B. Robinson,
*Black holes in quasi-topological gravity*,*JHEP***08**(2010) 067 [arXiv:1003.5357] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [45]J.T. Liu and W.A. Sabra,
*Hamilton-Jacobi counterterms for Einstein-Gauss-Bonnet gravity*,*Class. Quant. Grav.***27**(2010) 175014 [arXiv:0807.1256] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [46]Z.-Y. Fan, B. Chen and H. Lü,
*Criticality in Einstein-Gauss-Bonnet gravity: gravity without graviton*,*Eur. Phys. J.***C 76**(2016) 542 [arXiv:1606.02728] [INSPIRE]. - [47]S.C. Davis,
*Generalized Israel junction conditions for a Gauss-Bonnet brane world*,*Phys. Rev.***D 67**(2003) 024030 [hep-th/0208205] [INSPIRE]. - [48]H.-S. Liu, H. Lü and C.N. Pope,
*Holographic heat current as Noether current*,*JHEP***09**(2017) 146 [arXiv:1708.02329] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [49]Y.-Z. Li and H. Lü,
*a-theorem for Horndeski gravity at the critical point*,*Phys. Rev.***D 97**(2018) 126008 [arXiv:1803.08088] [INSPIRE]. - [50]G. Alkaç and B. Tekin,
*Holographic c-theorem and Born-Infeld gravity theories*,*Phys. Rev.***D 98**(2018) 046013 [arXiv:1805.07963] [INSPIRE]. - [51]N. Deruelle, M. Sasaki, Y. Sendouda and D. Yamauchi,
*Hamiltonian formulation of f(Riemann) theories of gravity*,*Prog. Theor. Phys.***123**(2010) 169 [arXiv:0908.0679] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [52]T.C. Sisman, I. Gullu and B. Tekin,
*All unitary cubic curvature gravities in D dimensions*,*Class. Quant. Grav.***28**(2011) 195004 [arXiv:1103.2307] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [53]J. Oliva and S. Ray,
*A new cubic theory of gravity in five dimensions: black hole, Birkhoff’s theorem and C-function*,*Class. Quant. Grav.***27**(2010) 225002 [arXiv:1003.4773] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [54]R.C. Myers, M.F. Paulos and A. Sinha,
*Holographic studies of quasi-topological gravity*,*JHEP***08**(2010) 035 [arXiv:1004.2055] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [55]P. Bueno, P.A. Cano and A. Ruipérez,
*Holographic studies of Einsteinian cubic gravity*,*JHEP***03**(2018) 150 [arXiv:1802.00018] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar