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Holographic Dual of Linear Dilaton Black Hole in Einstein-Maxwell-Dilaton-Axion Gravity
Authors
Ran li,Ji-Rong Ren
A. Joseph Layon,George Stainback,Stuart Young,Terrance Quinn II,Paolo Nason,Sunil Mukhi,Vladimir Makarenko,Salahaldin Fathalla,Niklas Beisert,Alessandro Paiella,Richard Jones,Andrzej Wal,Denis Comelli,Alberto Lerda,Зинаида Чеканцева,Vladimir Kulik,Andreas Hamann,Mikjel Thorsrud,DIGAMBER PAWAR,Trevor Glasbey,John Platt,Keisuke Yano,Kunal Vyas,Kristen Smith,Lawrence Kennedy,Nenad Đurđević,Luis Emilio Cruz-Barba,Frédéric Dreyer,Dejan Matić,Milivoje Lapčević,Veljko Vlašković,Bojan Urdarević,Zoran Vuković,Zoran Jovanović,Alexandru Ciubotaru,Emilija Stanković,Jelena Vučković,Ioannis Tsinikos,Tamara Đurđić Milošević,Srđan Vladetić,Jovan Vujičić,Sveto Puríć,Milan Palević,Peter Wolfsteiner,Nina Planojević,Milan Rapajić,Slavko Đorđević,Siniša Varga,Srdjan Đorđević,Torben Gaedt,Garrett Poe,Vesna Janjić,Jasmina Bogićević,Nada Todorović,Snežana Ljubisavljević,Vladimir Obradović,Dimitrios Zoakos,Dragan Stojković,Alain Jaccard,Carolyn Dzierba,Mirjana Todorović,Krešimir Kumerički,Biljana Jovković,Marc Torrens,Benjamin Bellenie,Stefan Hoeche,Stevan Luković,Srđan Furtula,Luis Garcés-Erice,Joseph Fotsing,Digamber Porob,David Van Wie,Milica Tufegdzic,Matti Jarvinen,Susana Henriques,N/A ,Jonathan Wall,Brian Lemoff,Marie Connett,Milan Čupić,Miljan Leković,Dejan Jovanović,Bulent Basol,Gideon Wolfaardt,Christopher Herzog,Martin Köhne,David Clarke,Andre Lukas,Mukund Rangamani,Dan Andersen,José Pedro Viegas Barros,Kim Landsbergen,Olli Pitkänen,CHIA-HO LIN,Mercedes Medina,Sebastian Linß,Sarah Kraak,Bruno Dufour,Omar Diego Lopez,Eran Palti,Nathaniel Butlin,Isabel María del Águila Cano,Sebastian Rosenbaum,Rafael Maestre,Michele Cicoli,Ingo Jordan,Luiz Agostinho Ferreira,Ivonne Zavala,Grigoris Panotopoulos,ALESSANDRO BROGGIO,Michael Taylor,Mark Begbie,Diosvany Ortega González,Yukiyoshi Okauchi,Gregory Carven,Stepan Jenicek,Suprabh Prakash,Leopoldo Pando Zayas,Yue-Lin Sming Tsai,Stuart Haigh,Nico Callewaert,Francesco Riva,Robert LoGalbo,Kevin O'Hara,Marinei Lopes Pedralli,Rong-Chi ,Stephane Coulombe,Brian Palik,Alexander Monin,Bruce Bassett,Gert Aarts,Hongzhi Luo,Rafel Escribano,Renee Hlozek,Louis Strigari,Jose Rubo,Gautam ,J. 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arXiv:1009.3139v1 [hep-th] 16 Sep 2010
Preprint typeset in JHEP style - HYPER VERSION
Holographic Dual of Linear Dilaton Black Hole in
Einstein-Maxwell-Dilaton-Axion Gravity
Ran Li
Institute of Modern Physics,
Chinese Academy of Sciences,
Lanzhou, 730000, Gansu, China
liran05@lzu.cn
Ji-Rong Ren
Institute of Theoretical Physics,
Lanzhou University,
Lanzhou, 730000, Gansu, China
renjr@lzu.edu.cn
Abstract: Motivated by the recently proposed Kerr/CFT correspondence, we investigate
the holographic dual of the extremal and non-extremal rotating linear dilaton black hole in
Einstein-Maxwell-Dilaton-Axion Gravity. For the case of extremal black hole, by imposing
the appropriate boundary condition at spatial infinity of the near horizon extremal geom-
etry, the Virasoro algebra of conserved charges associated with the asymptotic symmetry
group is obtained. It is shown that the microscopic entropy of the dual conformal field given
by Cardy formula exactly agrees with Bekenstein-Hawking entropy of extremal black hole.
Then, by rewriting the wave equation of massless scalar field with sufficient low energy as
the SL(2, R)L×SL(2, R)R Casimir operator, we find the hidden conformal symmetry of
the non-extremal linear dilaton black hole, which implies that the non-extremal rotating
linear dilaton black hole is holographically dual to a two dimensional conformal field theory
with the non-zero left and right temperatures. Furthermore, it is shown that the entropy
of non-extremal black hole can be reproduced by using Cardy formula.
Preprint typeset in JHEP style - HYPER VERSION
Holographic Dual of Linear Dilaton Black Hole in
Einstein-Maxwell-Dilaton-Axion Gravity
Ran Li
Institute of Modern Physics,
Chinese Academy of Sciences,
Lanzhou, 730000, Gansu, China
liran05@lzu.cn
Ji-Rong Ren
Institute of Theoretical Physics,
Lanzhou University,
Lanzhou, 730000, Gansu, China
renjr@lzu.edu.cn
Abstract: Motivated by the recently proposed Kerr/CFT correspondence, we investigate
the holographic dual of the extremal and non-extremal rotating linear dilaton black hole in
Einstein-Maxwell-Dilaton-Axion Gravity. For the case of extremal black hole, by imposing
the appropriate boundary condition at spatial infinity of the near horizon extremal geom-
etry, the Virasoro algebra of conserved charges associated with the asymptotic symmetry
group is obtained. It is shown that the microscopic entropy of the dual conformal field given
by Cardy formula exactly agrees with Bekenstein-Hawking entropy of extremal black hole.
Then, by rewriting the wave equation of massless scalar field with sufficient low energy as
the SL(2, R)L×SL(2, R)R Casimir operator, we find the hidden conformal symmetry of
the non-extremal linear dilaton black hole, which implies that the non-extremal rotating
linear dilaton black hole is holographically dual to a two dimensional conformal field theory
with the non-zero left and right temperatures. Furthermore, it is shown that the entropy
of non-extremal black hole can be reproduced by using Cardy formula.
Contents
1. Introduction 1
2. Rotating linear dilaton black hole 2
3. Holographic dual of extremal linear dilaton black hole 4
4. Hidden conformal symmetry of non-extremal linear dilaton black hole 7
5. Conclusion 11
1. Introduction
The recently proposed extremal Kerr/CFT correspondence [1] states that quantum grav-
ity in the region very near the event horizon of an extreme Kerr black hole with proper
boundary conditions is holographically dual to a two-dimensional chiral conformal field
theory with the central charge proportional to angular momentum. The method employed
by Guica, Hartman, Song and Strominger (GHSS) in [1] is very similar to the approach of
Brown and Henneaux in [2], where the AdS3 background is replaced by the near-horizon
extremal Kerr (NHEK) geometry previously obtained in [3]. They shown that, by im-
posing the appropriate boundary condition at spatial infinity of the NHEK geometry, the
conserved charges associated with the asymptotic symmetry group are found to form a copy
of Virasoro algebra. If identifying this algebra with the Virasoro algebra of the dual two
dimensional conformal field theory, it is shown that the macroscopic Bekenstein-Hawking
entropy of extremal Kerr black hole can be reproduced by the microscopic entropy of dual
conformal field theory via Cardy formula. This method has been generalized to calculate
the entropies of extremal black holes in a lot of theories such as the Einstein theory, string
theory, and supergravity theory. Some further studies on the extremal Kerr/CFT dual are
listed in [5]-[40].
More recently, Castro, Maloney and Strominger (CMS) in a remarkable paper [41]
show that there exists a hidden SL(2, R)L×SL(2, R)R conformal symmetry for the four
dimensional non-extremal Kerr black hole by studying the near-region wave equation of a
massless scalar field. Interestingly, this hidden conformal symmetry is not derived from the
conformal symmetry of spacetime geometry itself, but probed by the perturbation fields
in the near region. It is shown that, for the massive scalar field in the background of Kerr
black hole with sufficient low energy, the wave equation can be reproduced by the SL(2,
R)L×SL(2, R)R Casimir operator. It is also shown that microscopic entropy computed by
Cardy formula agrees exactly with the macroscopic Berenstein-Hawking entropy of non-
extremal Kerr black hole. These observations suggest that non-extremal Kerr black hole
– 1 –
1. Introduction 1
2. Rotating linear dilaton black hole 2
3. Holographic dual of extremal linear dilaton black hole 4
4. Hidden conformal symmetry of non-extremal linear dilaton black hole 7
5. Conclusion 11
1. Introduction
The recently proposed extremal Kerr/CFT correspondence [1] states that quantum grav-
ity in the region very near the event horizon of an extreme Kerr black hole with proper
boundary conditions is holographically dual to a two-dimensional chiral conformal field
theory with the central charge proportional to angular momentum. The method employed
by Guica, Hartman, Song and Strominger (GHSS) in [1] is very similar to the approach of
Brown and Henneaux in [2], where the AdS3 background is replaced by the near-horizon
extremal Kerr (NHEK) geometry previously obtained in [3]. They shown that, by im-
posing the appropriate boundary condition at spatial infinity of the NHEK geometry, the
conserved charges associated with the asymptotic symmetry group are found to form a copy
of Virasoro algebra. If identifying this algebra with the Virasoro algebra of the dual two
dimensional conformal field theory, it is shown that the macroscopic Bekenstein-Hawking
entropy of extremal Kerr black hole can be reproduced by the microscopic entropy of dual
conformal field theory via Cardy formula. This method has been generalized to calculate
the entropies of extremal black holes in a lot of theories such as the Einstein theory, string
theory, and supergravity theory. Some further studies on the extremal Kerr/CFT dual are
listed in [5]-[40].
More recently, Castro, Maloney and Strominger (CMS) in a remarkable paper [41]
show that there exists a hidden SL(2, R)L×SL(2, R)R conformal symmetry for the four
dimensional non-extremal Kerr black hole by studying the near-region wave equation of a
massless scalar field. Interestingly, this hidden conformal symmetry is not derived from the
conformal symmetry of spacetime geometry itself, but probed by the perturbation fields
in the near region. It is shown that, for the massive scalar field in the background of Kerr
black hole with sufficient low energy, the wave equation can be reproduced by the SL(2,
R)L×SL(2, R)R Casimir operator. It is also shown that microscopic entropy computed by
Cardy formula agrees exactly with the macroscopic Berenstein-Hawking entropy of non-
extremal Kerr black hole. These observations suggest that non-extremal Kerr black hole
– 1 –
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