Abstract
Recently, a practical approach to holographic renormalization has been developed based on the Hamilton-Jacobi formulation. Using a simple Einstein-scalar theory, we clarify that this approach does not conflict with the Hamiltonian constraint as it seems. Then we apply it to the holographic renormalization of massive gravity. We assume that the shift vector is falling off fast enough asymptotically. We derive the counterterms up to the boundary dimension d = 4. Interestingly, we find that the conformal anomaly can even occur in odd dimensions, which is different from the Einstein gravity. We check that the counterterms cancel the divergent part of the on-shell action at the background level. At the perturbation level, they are also applicable in several time-dependent cases.
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O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large N field theories, string theory and gravity, Phys. Rept.323 (2000) 183 [hep-th/9905111] [INSPIRE].
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].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys.2 (1998) 253 [hep-th/9802150] [INSPIRE].
I. Papadimitriou, Lectures on Holographic Renormalization, Springer Proc. Phys.176 (2016) 131 [INSPIRE].
M. Henningson and K. Skenderis, The Holographic Weyl anomaly, JHEP07 (1998) 023 [hep-th/9806087] [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. Bianchi, D.Z. Freedman and K. Skenderis, Holographic renormalization, Nucl. Phys.B 631 (2002) 159 [hep-th/0112119] [INSPIRE].
I. Papadimitriou and K. Skenderis, AdS/CFT correspondence and geometry, IRMA Lect. Math. Theor. Phys.8 (2005) 73 [hep-th/0404176] [INSPIRE].
C. Fefferman and C.R. Graham, Conformal invariants, in Elie Cartan et les Mathématiques d’aujour d’hui, AstériqueS131 (1985) 95.
J. de Boer, E.P. Verlinde and H.L. Verlinde, On the holographic renormalization group, JHEP08 (2000) 003 [hep-th/9912012] [INSPIRE].
J. de Boer, The Holographic renormalization group, Fortsch. Phys.49 (2001) 339 [hep-th/0101026] [INSPIRE].
D. Martelli and W. Mueck, Holographic renormalization and Ward identities with the Hamilton-Jacobi method, Nucl. Phys.B 654 (2003) 248 [hep-th/0205061] [INSPIRE].
J. Kalkkinen, D. Martelli and W. Mueck, Holographic renormalization and anomalies, JHEP04 (2001) 036 [hep-th/0103111] [INSPIRE].
I. Papadimitriou and K. Skenderis, Correlation functions in holographic RG flows, JHEP10 (2004) 075 [hep-th/0407071] [INSPIRE].
I. Papadimitriou, Holographic Renormalization of general dilaton-axion gravity, JHEP08 (2011) 119 [arXiv:1106.4826] [INSPIRE].
W. Chemissany and I. Papadimitriou, Generalized dilatation operator method for non-relativistic holography, Phys. Lett.B 737 (2014) 272 [arXiv:1405.3965] [INSPIRE].
W. Chemissany and I. Papadimitriou, Lifshitz holography: The whole shebang, JHEP01 (2015) 052 [arXiv:1408.0795] [INSPIRE].
J.D. Brown and J.W. York Jr., Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev.D 47 (1993) 1407 [gr-qc/9209012] [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. Olea, Mass, angular momentum and thermodynamics in four-dimensional Kerr-AdS black holes, JHEP06 (2005) 023 [hep-th/0504233] [INSPIRE].
R. Olea, Regularization of odd-dimensional AdS gravity: Kounterterms, JHEP04 (2007) 073 [hep-th/0610230] [INSPIRE].
A. Bzowski, Dimensional renormalization in AdS/CFT, arXiv:1612.03915 [INSPIRE].
H. Elvang and M. Hadjiantonis, A Practical Approach to the Hamilton-Jacobi Formulation of Holographic Renormalization, JHEP06 (2016) 046 [arXiv:1603.04485] [INSPIRE].
F. Larsen and R. McNees, Inflation and de Sitter holography, JHEP07 (2003) 051 [hep-th/0307026] [INSPIRE].
M. Fierz and W. Pauli, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field, Proc. Roy. Soc. Lond.A 173 (1939) 211 [INSPIRE].
H. van Dam and M.J.G. Veltman, Massive and massless Yang-Mills and gravitational fields, Nucl. Phys.B 22 (1970) 397 [INSPIRE].
V.I. Zakharov, Linearized gravitation theory and the graviton mass, JETP Lett.12 (1970) 312 [INSPIRE].
A.I. Vainshtein, To the problem of nonvanishing gravitation mass, Phys. Lett.39B (1972) 393 [INSPIRE].
D.G. Boulware and S. Deser, Can gravitation have a finite range?, Phys. Rev.D 6 (1972) 3368 [INSPIRE].
K. Hinterbichler, Theoretical Aspects of Massive Gravity, Rev. Mod. Phys.84 (2012) 671 [arXiv:1105.3735] [INSPIRE].
C. de Rham and G. Gabadadze, Generalization of the Fierz-Pauli Action, Phys. Rev.D 82 (2010) 044020 [arXiv:1007.0443] [INSPIRE].
C. de Rham, G. Gabadadze and A.J. Tolley, Resummation of Massive Gravity, Phys. Rev. Lett.106 (2011) 231101 [arXiv:1011.1232] [INSPIRE].
C. de Rham, Massive Gravity, Living Rev. Rel.17 (2014) 7 [arXiv:1401.4173] [INSPIRE].
D. Vegh, Holography without translational symmetry, arXiv:1301.0537 [INSPIRE].
R.A. Davison, Momentum relaxation in holographic massive gravity, Phys. Rev.D 88 (2013) 086003 [arXiv:1306.5792] [INSPIRE].
M. Blake and D. Tong, Universal Resistivity from Holographic Massive Gravity, Phys. Rev.D 88 (2013) 106004 [arXiv:1308.4970] [INSPIRE].
M. Blake, D. Tong and D. Vegh, Holographic Lattices Give the Graviton an Effective Mass, Phys. Rev. Lett.112 (2014) 071602 [arXiv:1310.3832] [INSPIRE].
A. Amoretti, A. Braggio, N. Maggiore, N. Magnoli and D. Musso, Thermo-electric transport in gauge/gravity models with momentum dissipation, JHEP09 (2014) 160 [arXiv:1406.4134] [INSPIRE].
M. Baggioli and O. Pujolàs, Electron-Phonon Interactions, Metal-Insulator Transitions and Holographic Massive Gravity, Phys. Rev. Lett.114 (2015) 251602 [arXiv:1411.1003] [INSPIRE].
L.-M. Cao and Y. Peng, Counterterms in Massive Gravity Theory, Phys. Rev.D 92 (2015) 124052 [arXiv:1509.08738] [INSPIRE].
L.D. Landau and E.M. Lifshitz, Mechanics, Course of Theoretical Physics, Vol. 1: Mechanics, Pergamon Press, London (1987).
B.P. Dolan, Symplectic geometry and Hamiltonian flow of the renormalization group equation, Int. J. Mod. Phys.A 10 (1995) 2703 [hep-th/9406061] [INSPIRE].
I. Papadimitriou, Holographic renormalization as a canonical transformation, JHEP11 (2010) 014 [arXiv:1007.4592] [INSPIRE].
S.M. Carroll, Spacetime and Geometry: An Introduction to General Relativity, Addison Wesley, San Francisco (2004).
R.-G. Cai, Y.-P. Hu, Q.-Y. Pan and Y.-L. Zhang, Thermodynamics of Black Holes in Massive Gravity, Phys. Rev.D 91 (2015) 024032 [arXiv:1409.2369] [INSPIRE].
P. Breitenlohner and D.Z. Freedman, Positive Energy in anti-de Sitter Backgrounds and Gauged Extended Supergravity, Phys. Lett.115B (1982) 197 [INSPIRE].
H. Goldstein, C. Poole and J. Safko, Classical mechanics, Addison Wesley (2001).
K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav.19 (2002) 5849 [hep-th/0209067] [INSPIRE].
L.-M. Cao, Y. Peng and Y.-L. Zhang, de Rham-Gabadadze-Tolley massive gravity with degenerate reference metrics, Phys. Rev.D 93 (2016) 124015 [arXiv:1511.04967] [INSPIRE].
L. Bernard, C. Deffayet and M. von Strauss, Massive graviton on arbitrary background: derivation, syzygies, applications, JCAP06 (2015) 038 [arXiv:1504.04382] [INSPIRE].
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Chen, F., Wu, SF. & Peng, Y. Hamilton-Jacobi approach to holographic renormalization of massive gravity. J. High Energ. Phys. 2019, 72 (2019). https://doi.org/10.1007/JHEP07(2019)072
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DOI: https://doi.org/10.1007/JHEP07(2019)072