# AdS_{2} holographic dictionary

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## Abstract

We construct the holographic dictionary for both running and constant dilaton solutions of the two dimensional Einstein-Maxwell-Dilaton theory that is obtained by a circle reduction from Einstein-Hilbert gravity with negative cosmological constant in three dimensions. This specific model ensures that the dual theory has a well defined ultraviolet completion in terms of a two dimensional conformal field theory, but our results apply qualitatively to a wider class of two dimensional dilaton gravity theories. For each type of solutions we perform holographic renormalization, compute the exact renormalized one-point functions in the presence of arbitrary sources, and derive the asymptotic symmetries and the corresponding conserved charges. In both cases we find that the scalar operator dual to the dilaton plays a crucial role in the description of the dynamics. Its source gives rise to a matter conformal anomaly for the running dilaton solutions, while its expectation value is the only non trivial observable for constant dilaton solutions. The role of this operator has been largely overlooked in the literature. We further show that the only non trivial conserved charges for running dilaton solutions are the mass and the electric charge, while for constant dilaton solutions only the electric charge is non zero. However, by uplifting the solutions to three dimensions we show that constant dilaton solutions can support non trivial extended symmetry algebras, including the one found by Compère, Song and Strominger [1], in agreement with the results of Castro and Song [2]. Finally, we demonstrate that any solution of this specific dilaton gravity model can be uplifted to a family of asymptotically AdS_{2} × *S* ^{2} or conformally AdS_{2} × *S* ^{2} solutions of the STU model in four dimensions, including non extremal black holes. The four dimensional solutions obtained by uplifting the running dilaton solutions coincide with the so called ‘subtracted geometries’, while those obtained from the uplift of the constant dilaton ones are new.

## Keywords

2D Gravity AdS-CFT Correspondence Black Holes Space-Time Symmetries## Notes

### **Open Access**

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## References

- [1]G. Compère, W. Song and A. Strominger,
*New boundary conditions for AdS*_{3},*JHEP***05**(2013) 152 [arXiv:1303.2662] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [2]
- [3]T. Hartman and A. Strominger,
*Central charge for AdS*_{2}*quantum gravity*,*JHEP***04**(2009) 026 [arXiv:0803.3621] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [4]A. Strominger,
*AdS*_{2}*quantum gravity and string theory*,*JHEP***01**(1999) 007 [hep-th/9809027] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [5]A. Chamblin, R. Emparan, C.V. Johnson and R.C. Myers,
*Charged AdS black holes and catastrophic holography*,*Phys. Rev.***D 60**(1999) 064018 [hep-th/9902170] [INSPIRE].ADSMathSciNetGoogle Scholar - [6]J.M. Maldacena, J. Michelson and A. Strominger,
*Anti-de Sitter fragmentation*,*JHEP***02**(1999) 011 [hep-th/9812073] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [7]M. Cadoni and S. Mignemi,
*Asymptotic symmetries of AdS*_{2}*and conformal group in D*= 1,*Nucl. Phys.***B 557**(1999) 165 [hep-th/9902040] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [8]M. Spradlin and A. Strominger,
*Vacuum states for AdS*_{2}*black holes*,*JHEP***11**(1999) 021 [hep-th/9904143] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [9]J. Navarro-Salas and P. Navarro,
*AdS*_{2}*/CFT*_{1}*correspondence and near extremal black hole entropy*,*Nucl. Phys.***B 579**(2000) 250 [hep-th/9910076] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [10]M. Cadoni and S. Mignemi,
*Symmetry breaking, central charges and the AdS*_{2}*/CFT*_{1}*correspondence*,*Phys. Lett.***B 490**(2000) 131 [hep-th/0002256] [INSPIRE].ADSCrossRefGoogle Scholar - [11]M. Caldarelli, G. Catelani and L. Vanzo,
*Action, Hamiltonian and CFT for*2*D black holes*,*JHEP***10**(2000) 005 [hep-th/0008058] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [12]M. Cadoni, P. Carta, D. Klemm and S. Mignemi,
*AdS*_{2}*gravity as conformally invariant mechanical system*,*Phys. Rev.***D 63**(2001) 125021 [hep-th/0009185] [INSPIRE].ADSGoogle Scholar - [13]M. Alishahiha and F. Ardalan,
*Central charge for*2*D gravity on AdS*_{2}*and AdS*_{2}*/CFT*_{1}*correspondence*,*JHEP***08**(2008) 079 [arXiv:0805.1861] [INSPIRE].ADSCrossRefGoogle Scholar - [14]A. Castro, D. Grumiller, F. Larsen and R. McNees,
*Holographic description of AdS*_{2}*black holes*,*JHEP***11**(2008) 052 [arXiv:0809.4264] [INSPIRE].ADSCrossRefGoogle Scholar - [15]M. Alishahiha, R. Fareghbal and A.E. Mosaffa, 2
*D gravity on AdS*_{2}*with Chern-Simons corrections*,*JHEP***01**(2009) 069 [arXiv:0812.0453] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [16]D. Grumiller, M. Leston and D. Vassilevich,
*Anti-de Sitter holography for gravity and higher spin theories in two dimensions*,*Phys. Rev.***D 89**(2014) 044001 [arXiv:1311.7413] [INSPIRE].ADSGoogle Scholar - [17]A. Almheiri and J. Polchinski,
*Models of AdS*_{2}*backreaction and holography*,*JHEP***11**(2015) 014 [arXiv:1402.6334] [INSPIRE].ADSCrossRefGoogle Scholar - [18]D. Grumiller, J. Salzer and D. Vassilevich,
*AdS*_{2}*holography is (non-)trivial for (non-)constant dilaton*,*JHEP***12**(2015) 015 [arXiv:1509.08486] [INSPIRE].ADSGoogle Scholar - [19]K. Jensen,
*Chaos in AdS*_{2}*holography*,*Phys. Rev. Lett.***117**(2016) 111601 [arXiv:1605.06098] [INSPIRE].ADSCrossRefGoogle Scholar - [20]J. Maldacena, D. Stanford and Z. Yang,
*Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space*, arXiv:1606.01857 [INSPIRE]. - [21]J. Engelsöy, T.G. Mertens and H. Verlinde,
*An investigation of AdS*_{2}*backreaction and holography*,*JHEP***07**(2016) 139 [arXiv:1606.03438] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [22]A. Sen,
*Entropy Function and AdS*_{2}*/CFT*_{1}*Correspondence*,*JHEP***11**(2008) 075 [arXiv:0805.0095] [INSPIRE].ADSCrossRefGoogle Scholar - [23]R.K. Gupta and A. Sen,
*AdS*_{3}*/CFT*_{2}*to AdS*_{2}*/CFT*_{1},*JHEP***04**(2009) 034 [arXiv:0806.0053] [INSPIRE].ADSCrossRefGoogle Scholar - [24]A. Sen,
*Quantum entropy function from AdS*_{2}*/CFT*_{1}*correspondence*,*Int. J. Mod. Phys.***A 24**(2009) 4225 [arXiv:0809.3304] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [25]S. Sachdev,
*Holographic metals and the fractionalized Fermi liquid*,*Phys. Rev. Lett.***105**(2010) 151602 [arXiv:1006.3794] [INSPIRE].ADSCrossRefGoogle Scholar - [26]J. Maldacena and D. Stanford,
*Remarks on the Sachdev-Ye-Kitaev model*,*Phys. Rev.***D 94**(2016) 106002 [arXiv:1604.07818] [INSPIRE].ADSGoogle Scholar - [27]S. Sachdev and J. Ye,
*Gapless spin-fluid ground state in a random quantum Heisenberg magnet*,*Phys. Rev. Lett.***70**(1993) 3339 [cond-mat/9212030]. - [28]
- [29]J. Erdmenger, C. Hoyos, A. O’Bannon and J. Wu,
*A holographic model of the Kondo effect*,*JHEP***12**(2013) 086 [arXiv:1310.3271] [INSPIRE].ADSCrossRefGoogle Scholar - [30]A. Almheiri and B. Kang,
*Conformal symmetry breaking and thermodynamics of near-extremal black holes*,*JHEP***10**(2016) 052 [arXiv:1606.04108] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [31]R. Jackiw,
*Liouville field theory: A two-dimensional model for gravity*, MIT-CTP-1049 (1982) [INSPIRE]. - [32]C. Teitelboim,
*The Hamiltonian structure of two-dimensional space-time and its relation with the conformal anomaly*(1983).Google Scholar - [33]M. Cvetič and F. Larsen,
*Conformal symmetry for general black holes*,*JHEP***02**(2012) 122 [arXiv:1106.3341] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [34]M. Cvetič and F. Larsen,
*Conformal symmetry for black holes in four dimensions*,*JHEP***09**(2012) 076 [arXiv:1112.4846] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [35]M. Cvetič and G.W. Gibbons,
*Conformal symmetry of a black hole as a scaling limit: a black hole in an asymptotically conical box*,*JHEP***07**(2012) 014 [arXiv:1201.0601] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [36]A. Virmani,
*Subtracted geometry from harrison transformations*,*JHEP***07**(2012) 086 [arXiv:1203.5088] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [37]M. Baggio, J. de Boer, J.I. Jottar and D.R. Mayerson,
*Conformal symmetry for black holes in four dimensions and irrelevant deformations*,*JHEP***04**(2013) 084 [arXiv:1210.7695] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [38]M. Cvetič, M. Guica and Z.H. Saleem,
*General black holes, untwisted*,*JHEP***09**(2013) 017 [arXiv:1302.7032] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [39]M. Cvetič and F. Larsen,
*Black holes with intrinsic spin*,*JHEP***11**(2014) 033 [arXiv:1406.4536] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [40]M. Cvetič and D. Youm,
*All the static spherically symmetric black holes of heterotic string on a six torus*,*Nucl. Phys.***B 472**(1996) 249 [hep-th/9512127] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar - [41]M. Cvetič and D. Youm,
*Entropy of nonextreme charged rotating black holes in string theory*,*Phys. Rev.***D 54**(1996) 2612 [hep-th/9603147] [INSPIRE].ADSMathSciNetGoogle Scholar - [42]Z.W. Chong, M. Cvetič, H. Lü and C.N. Pope,
*Charged rotating black holes in four-dimensional gauged and ungauged supergravities*,*Nucl. Phys.***B 717**(2005) 246 [hep-th/0411045] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [43]D.D.K. Chow and G. Compère,
*Seed for general rotating non-extremal black holes of*\( \mathcal{N} \) = 8*supergravity*,*Class. Quant. Grav.***31**(2014) 022001 [arXiv:1310.1925] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [44]D.D.K. Chow and G. Compère,
*Black holes in N*= 8*supergravity from*SO(4*,*4)*hidden symmetries*,*Phys. Rev.***D 90**(2014) 025029 [arXiv:1404.2602] [INSPIRE].ADSGoogle Scholar - [45]M. Cvetič and D. Youm,
*General rotating five-dimensional black holes of toroidally compactified heterotic string*,*Nucl. Phys.***B 476**(1996) 118 [hep-th/9603100] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [46]M. Cvetič, Z.H. Saleem and A. Satz,
*Entanglement entropy of subtracted geometry black holes*,*JHEP***09**(2014) 041 [*Erratum ibid.***09**(2015) 099] [arXiv:1407.0310] [INSPIRE]. - [47]M. Cvetič, G.W. Gibbons, Z.H. Saleem and A. Satz,
*Vacuum polarization of STU black holes and their subtracted geometry limit*,*JHEP***01**(2015) 130 [arXiv:1411.4658] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [48]M. Cvetič, Z.H. Saleem and A. Satz,
*Analytical result for the vacuum polarization of subtracted rotating black holes*,*Phys. Rev.***D 92**(2015) 064030 [arXiv:1506.07189] [INSPIRE].ADSMathSciNetGoogle Scholar - [49]N. Itzhaki, J.M. Maldacena, J. Sonnenschein and S. Yankielowicz,
*Supergravity and the large-N limit of theories with sixteen supercharges*,*Phys. Rev.***D 58**(1998) 046004 [hep-th/9802042] [INSPIRE].ADSMathSciNetGoogle Scholar - [50]I. Kanitscheider, K. Skenderis and M. Taylor,
*Precision holography for non-conformal branes*,*JHEP***09**(2008) 094 [arXiv:0807.3324] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [51]O.S. An, M. Cvetič and I. Papadimitriou,
*Black hole thermodynamics from a variational principle: Asymptotically conical backgrounds*,*JHEP***03**(2016) 086 [arXiv:1602.01508] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [52]J.M. Bardeen and G.T. Horowitz,
*The extreme Kerr throat geometry: a vacuum analog of AdS*_{2}×*S*^{2},*Phys. Rev.***D 60**(1999) 104030 [hep-th/9905099] [INSPIRE].ADSMathSciNetGoogle Scholar - [53]J. Erdmenger, C. Hoyos, A. O’Bannon, I. Papadimitriou, J. Probst and J. Wu,
*Two-point functions in a holographic Kondo model*, in preparation.Google Scholar - [54]A.D. Polyanin and V.F. Zaitsev,
*Handbook of exact solutions for ordinary differential equations*, CRC Press, U.S.A. (2003).zbMATHGoogle Scholar - [55]R.M. Wald,
*Black hole entropy is the Noether charge*,*Phys. Rev.***D 48**(1993) R3427 [gr-qc/9307038] [INSPIRE]. - [56]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]. - [57]R.C. Myers,
*Black hole entropy in two-dimensions*,*Phys. Rev.***D 50**(1994) 6412 [hep-th/9405162] [INSPIRE].ADSMathSciNetGoogle Scholar - [58]J. Gegenberg, G. Kunstatter and D. Louis-Martinez,
*Observables for two-dimensional black holes*,*Phys. Rev.***D 51**(1995) 1781 [gr-qc/9408015] [INSPIRE]. - [59]J.D. Brown and M. Henneaux,
*Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity*,*Commun. Math. Phys.***104**(1986) 207 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [60]I. Papadimitriou and K. Skenderis,
*AdS boundary conditions and black hole thermodynamics: scalars, antisymmetric tensors, and topological charges*, in preparation.Google Scholar - [61]V. Balasubramanian, J. de Boer, M.M. Sheikh-Jabbari and J. Simon,
*What is a chiral*2*D CFT? And what does it have to do with extremal black holes?*,*JHEP***02**(2010) 017 [arXiv:0906.3272] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [62]S. de Haro, I. Papadimitriou and A.C. Petkou,
*Conformally coupled scalars, instantons and vacuum instability in AdS*_{4},*Phys. Rev. Lett.***98**(2007) 231601 [hep-th/0611315] [INSPIRE].ADSCrossRefGoogle Scholar - [63]I. Papadimitriou,
*Multi-trace deformations in AdS/CFT: exploring the vacuum structure of the deformed CFT*,*JHEP***05**(2007) 075 [hep-th/0703152] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [64]I. Papadimitriou,
*Holographic renormalization of general dilaton-axion gravity*,*JHEP***08**(2011) 119 [arXiv:1106.4826] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [65]M. Henningson and K. Skenderis,
*The holographic weyl anomaly*,*JHEP***07**(1998) 023 [hep-th/9806087] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [66]I. Papadimitriou and K. Skenderis,
*AdS/CFT correspondence and geometry*,*IRMA Lect. Math. Theor. Phys.***8**(2005) 73 [hep-th/0404176] [INSPIRE].MathSciNetzbMATHGoogle Scholar - [67]H. Osborn,
*Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories*,*Nucl. Phys.***B 363**(1991) 486 [INSPIRE].ADSCrossRefGoogle Scholar - [68]A. O’Bannon, I. Papadimitriou and J. Probst,
*A holographic two-impurity Kondo model*,*JHEP***01**(2016) 103 [arXiv:1510.08123] [INSPIRE].MathSciNetCrossRefGoogle Scholar - [69]I. Papadimitriou and K. Skenderis,
*Thermodynamics of asymptotically locally AdS spacetimes*,*JHEP***08**(2005) 004 [hep-th/0505190] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [70]I. Papadimitriou,
*Holographic renormalization as a canonical transformation*,*JHEP***11**(2010) 014 [arXiv:1007.4592] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [71]W. Chemissany and I. Papadimitriou,
*Lifshitz holography: the whole shebang*,*JHEP***01**(2015) 052 [arXiv:1408.0795] [INSPIRE].ADSCrossRefGoogle Scholar - [72]D. Grumiller, R. McNees and J. Salzer,
*Cosmological constant as confining*U(1)*charge in two-dimensional dilaton gravity*,*Phys. Rev.***D 90**(2014) 044032 [arXiv:1406.7007] [INSPIRE].ADSGoogle Scholar - [73]K. Skenderis and S.N. Solodukhin,
*Quantum effective action from the AdS/CFT correspondence*,*Phys. Lett.***B 472**(2000) 316 [hep-th/9910023] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [74]C. Troessaert,
*Enhanced asymptotic symmetry algebra of AdS*_{3},*JHEP***08**(2013) 044 [arXiv:1303.3296] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar - [75]R. Penrose and W. Rindler,
*Spinors and spacetime*, volume 2, Cambridge Universiyt Press (1986), see chapter 9.Google Scholar - [76]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 - [77]S.G. Avery, R.R. Poojary and N.V. Suryanarayana,
*An*\( \mathrm{S}\mathrm{L}\left(2,\mathbb{R}\right) \)*current algebra from AdS*_{3}*gravity*,*JHEP***01**(2014) 144 [arXiv:1304.4252] [INSPIRE].ADSCrossRefGoogle Scholar