Abstract
We present a streamlined proof that any Einstein-AdS space is a solution of the Lu, Pang and Pope conformal gravity theory in six dimensions. The reduction of conformal gravity into Einstein theory manifestly shows that the action of the latter can be written as the Einstein-Hilbert term plus the Euler topological density and an additional contribution that depends on the Laplacian of the bulk Weyl tensor squared. The prescription for obtaining this form of the action by embedding the Einstein theory into a Weyl-invariant purely metric theory, was dubbed Conformal Renormalization and its resulting action was shown to be equivalent to the one obtained by holographic renormalization. As a non-trivial application of the method, we compute the Noether-Wald charges and thermodynamic quantities for topological black hole solutions with generic transverse section in Einstein-AdS6 theory.
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References
H. Weyl, Gravitation and electricity, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1918 (1918) 465 [INSPIRE].
H. Weyl, A New Extension of Relativity Theory, Annalen Phys. 59 (1919) 101 [INSPIRE].
R. Bach, Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungstensorbegriffs, Math. Z. 9 (1921) 110.
K.S. Stelle, Renormalization of Higher Derivative Quantum Gravity, Phys. Rev. D 16 (1977) 953 [INSPIRE].
D.M. Capper and M.J. Duff, Conformal Anomalies and the Renormalizability Problem in Quantum Gravity, Phys. Lett. A 53 (1975) 361 [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, Renormalizable asymptotically free quantum theory of gravity, Nucl. Phys. B 201 (1982) 469 [INSPIRE].
J. Julve and M. Tonin, Quantum Gravity with Higher Derivative Terms, Nuovo Cim. B 46 (1978) 137 [INSPIRE].
P.D. Mannheim and D. Kazanas, Exact Vacuum Solution to Conformal Weyl Gravity and Galactic Rotation Curves, Astrophys. J. 342 (1989) 635 [INSPIRE].
P.D. Mannheim, Alternatives to dark matter and dark energy, Prog. Part. Nucl. Phys. 56 (2006) 340 [astro-ph/0505266] [INSPIRE].
P.D. Mannheim and J.G. O’Brien, Impact of a global quadratic potential on galactic rotation curves, Phys. Rev. Lett. 106 (2011) 121101 [arXiv:1007.0970] [INSPIRE].
P.D. Mannheim, Making the Case for Conformal Gravity, Found. Phys. 42 (2012) 388 [arXiv:1101.2186] [INSPIRE].
N. Berkovits and E. Witten, Conformal supergravity in twistor-string theory, JHEP 08 (2004) 009 [hep-th/0406051] [INSPIRE].
M. Kaku and P.K. Townsend, Poincaré supergravity as broken superconformal gravity, Phys. Lett. B 76 (1978) 54 [INSPIRE].
M. Kaku, P.K. Townsend and P. van Nieuwenhuizen, Properties of Conformal Supergravity, Phys. Rev. D 17 (1978) 3179 [INSPIRE].
M. Kaku, P.K. Townsend and P. van Nieuwenhuizen, Gauge Theory of the Conformal and Superconformal Group, Phys. Lett. B 69 (1977) 304 [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, Conformal Supergravity, Phys. Rept. 119 (1985) 233 [INSPIRE].
B. de Wit, J.W. van Holten and A. Van Proeyen, Structure of N = 2 Supergravity, Nucl. Phys. B 184 (1981) 77 [Erratum ibid. 222 (1983) 516] [INSPIRE].
E. Bergshoeff, M. de Roo and B. de Wit, Extended Conformal Supergravity, Nucl. Phys. B 182 (1981) 173 [INSPIRE].
H. Liu and A.A. Tseytlin, D = 4 superYang-Mills, D = 5 gauged supergravity, and D = 4 conformal supergravity, Nucl. Phys. B 533 (1998) 88 [hep-th/9804083] [INSPIRE].
S. Ferrara, A. Kehagias and D. Lüst, Aspects of Weyl Supergravity, JHEP 08 (2018) 197 [arXiv:1806.10016] [INSPIRE].
L. Andrianopoli and R. D’Auria, N = 1 and N = 2 pure supergravities on a manifold with boundary, JHEP 08 (2014) 012 [arXiv:1405.2010] [INSPIRE].
R. D’Auria and L. Ravera, Conformal gravity with totally antisymmetric torsion, Phys. Rev. D 104 (2021) 084034 [arXiv:2101.10978] [INSPIRE].
S. Ferrara, A. Kehagias and D. Lüst, Aspects of Conformal Supergravity, in the proceedings of the 57th International School of Subnuclear Physics: In Search for the Unexpected, Erice Italy, 21–30 June (2019) [arXiv:2001.04998] [INSPIRE].
A.H. Chamseddine and A. Connes, The Spectral action principle, Commun. Math. Phys. 186 (1997) 731 [hep-th/9606001] [INSPIRE].
G. Manolakos, P. Manousselis and G. Zoupanos, Four-dimensional Gravity on a Covariant Noncommutative Space, JHEP 08 (2020) 001 [arXiv:1902.10922] [INSPIRE].
G. Manolakos, P. Manousselis and G. Zoupanos, Four-Dimensional Gravity on a Covariant Noncommutative Space (II), Fortsch. Phys. 69 (2021) 2100085 [arXiv:2104.13746] [INSPIRE].
M. Ostrogradsky, Mémoires sur les équations différentielles, relatives au problème des isopérimètres, Mem. Acad. St. Petersbourg 6 (1850) 385 [INSPIRE].
R.J. Riegert, The particle content of linearized conformal gravity, Phys. Lett. A 105 (1984) 110 [INSPIRE].
P.D. Mannheim, Solution to the ghost problem in higher-derivative gravity, Nuovo Cim. C 45 (2022) 27 [arXiv:2109.12743] [INSPIRE].
J. Maldacena, Einstein Gravity from Conformal Gravity, arXiv:1105.5632 [INSPIRE].
D. Grumiller, M. Irakleidou, I. Lovrekovic and R. McNees, Conformal gravity holography in four dimensions, Phys. Rev. Lett. 112 (2014) 111102 [arXiv:1310.0819] [INSPIRE].
C. Fefferman and C.R. Graham, Conformal invariants, Astérisque S131 (1985) 95 [http://www.numdam.org/item/AST_1985_S131_95_0].
A. Hell, D. Lüst and G. Zoupanos, On the ghost problem of conformal gravity, JHEP 08 (2023) 168 [arXiv:2306.13714] [INSPIRE].
G. Anastasiou and R. Olea, From conformal to Einstein Gravity, Phys. Rev. D 94 (2016) 086008 [arXiv:1608.07826] [INSPIRE].
S.W. MacDowell and F. Mansouri, Unified Geometric Theory of Gravity and Supergravity, Phys. Rev. Lett. 38 (1977) 739 [Erratum ibid. 38 (1977) 1376] [INSPIRE].
O. Miskovic and R. Olea, Topological regularization and self-duality in four-dimensional anti-de Sitter gravity, Phys. Rev. D 79 (2009) 124020 [arXiv:0902.2082] [INSPIRE].
G. Anastasiou, I.J. Araya and R. Olea, Einstein Gravity from Conformal Gravity in 6D, JHEP 01 (2021) 134 [arXiv:2010.15146] [INSPIRE].
H. Lü, Y. Pang and C.N. Pope, Black Holes in Six-dimensional Conformal Gravity, Phys. Rev. D 87 (2013) 104013 [arXiv:1301.7083] [INSPIRE].
R. Olea, Mass, angular momentum and thermodynamics in four-dimensional Kerr-AdS black holes, JHEP 06 (2005) 023 [hep-th/0504233] [INSPIRE].
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].
M. Henningson and K. Skenderis, The Holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].
G. Anastasiou et al., Conformal renormalization of scalar-tensor theories, Phys. Rev. D 107 (2023) 104049 [arXiv:2212.04364] [INSPIRE].
G. Anastasiou, I.J. Araya, C. Arias and R. Olea, Einstein-AdS action, renormalized volume/area and holographic Rényi entropies, JHEP 08 (2018) 136 [arXiv:1806.10708] [INSPIRE].
G. Anastasiou, I.J. Araya, A. Guijosa and R. Olea, Renormalized AdS gravity and holographic entanglement entropy of even-dimensional CFTs, JHEP 10 (2019) 221 [arXiv:1908.11447] [INSPIRE].
G. Anastasiou, O. Miskovic, R. Olea and I. Papadimitriou, Counterterms, Kounterterms, and the variational problem in AdS gravity, JHEP 08 (2020) 061 [arXiv:2003.06425] [INSPIRE].
I.J. Araya et al., Universal renormalization procedure for higher curvature gravities in D ≤ 5, JHEP 09 (2021) 142 [arXiv:2108.01126] [INSPIRE].
T. Eguchi, P.B. Gilkey and A.J. Hanson, Gravitation, Gauge Theories and Differential Geometry, Phys. Rept. 66 (1980) 213 [INSPIRE].
K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].
I. Papadimitriou and K. Skenderis, AdS/CFT correspondence and geometry, IRMA Lect. Math. Theor. Phys. 8 (2005) 73 [hep-th/0404176] [INSPIRE].
V. Balasubramanian and P. Kraus, A Stress tensor for Anti-de Sitter gravity, Commun. Math. Phys. 208 (1999) 413 [hep-th/9902121] [INSPIRE].
R. Emparan, C.V. Johnson and R.C. Myers, Surface terms as counterterms in the AdS/CFT correspondence, Phys. Rev. D 60 (1999) 104001 [hep-th/9903238] [INSPIRE].
V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
G. Anastasiou, I.J. Araya, C. Corral and R. Olea, Noether-Wald charges in six-dimensional Critical Gravity, JHEP 07 (2021) 156 [arXiv:2105.02924] [INSPIRE].
H. Osborn and A. Stergiou, Structures on the Conformal Manifold in Six Dimensional Theories, JHEP 04 (2015) 157 [arXiv:1501.01308] [INSPIRE].
Acknowledgments
We thank Gastón Giribet for comments and discussions. The work of GA is funded by ANID, Convocatoria Nacional Subvención a Instalación en la Academia Convocatoria Año 2021, Folio SA77210007. IJA is supported by ANID FONDECYT grants No. 11230419 and 1231133, and by ANID Becas Chile grant No. 74220042. IJA also acknowledges funding by ANID, REC Convocatoria Nacional Subvención a Instalación en la Academia Convocatoria Año 2020, Folio PAI77200097. IJA is grateful to Andrei Parnachev and the School of Mathematics at Trinity College Dublin for their hospitality. CC is partially supported by Agencia Nacional de Investigación y Desarrollo (ANID) through FONDECYT grants No 11200025, 1230112, and 1210500. The work of RO has been funded by the FONDECYT Regular Grants 1230492 and 1231779, and ANILLO Grant ANID/ACT210100.
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Anastasiou, G., Araya, I.J., Corral, C. et al. Conformal Renormalization of topological black holes in AdS6. J. High Energ. Phys. 2023, 36 (2023). https://doi.org/10.1007/JHEP11(2023)036
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DOI: https://doi.org/10.1007/JHEP11(2023)036