Abstract
We investigate aspects of holographic duals to time-like warped AdS 3 space-times — which include Gödel’s universe — in string theory. Using worldsheet techniques similar to those that have been applied to AdS 3 backgrounds, we are able to identify space-time symmetry algebras that act on the dual boundary theory. In particular, we always find at least one Virasoro algebra with computable central charge. Interestingly, there exists a dense set of points in the moduli space of these models in which there is actually a second commuting Virasoro algebra, typically with different central charge than the first. We analyze the supersymmetry of the backgrounds, finding related enhancements, and comment on possible interpretations of these results. We also perform an asymptotic symmetry analysis at the level of supergravity, providing additional support for the world sheet analysis.
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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 [SPIRES].
D. Birmingham, I. Sachs and S.N. Solodukhin, Conformal field theory interpretation of black hole quasi-normal modes, Phys. Rev. Lett. 88 (2002) 151301 [hep-th/0112055] [SPIRES].
S. Carlip, Conformal field theory, (2+ 1)-dimensional gravity and the BTZ black hole, Class. Quant. Grav. 22 (2005) R85 [gr-qc/0503022] [SPIRES].
A. Maloney and E. Witten, Quantum gravity partition functions in three dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [SPIRES].
M.R. Gaberdiel, Constraints on extremal self-dual CFTs, JHEP 11 (2007) 087 [arXiv:0707.4073] [SPIRES].
M. Guica, T. Hartman, W. Song and A. Strominger, The Kerr/CFT correspondence, Phys. Rev. D 80 (2009) 124008 [arXiv:0809.4266] [SPIRES].
S.A. Hartnoll, Lectures on holographic methods for condensed matter physics, Class. Quant. Grav. 26 (2009) 224002 [arXiv:0903.3246] [SPIRES].
S.S. Gubser and A. Karch, From gauge-string duality to strong interactions: a pedestrian’s guide, Ann. Rev. Nucl. Part. Sci. 59 (2009) 145 [arXiv:0901.0935] [SPIRES].
K. Godel, An Example of a new type of cosmological solutions of Einstein’s field equations of graviation, Rev. Mod. Phys. 21 (1949) 447 [SPIRES].
M.J. Reboucas and J. Tiomno, On the homogeneity of Riemannian space-times of Gödel type, Phys. Rev. D 28 (1983) 1251 [SPIRES].
M. Rooman and P. Spindel, Gödel metric as a squashed Anti-de Sitter geometry, Class. Quant. Grav. 15 (1998) 3241 [gr-qc/9804027] [SPIRES].
B. Chen, B. Ning and Z.-b. Xu, Real-time correlators in warped AdS/CFT correspondence, JHEP 02 (2010) 031 [arXiv:0911.0167] [SPIRES].
K.S. Gupta, E. Harikumar, S. Sen and M. Sivakumar, Geometric finiteness, holography and quasinormal modes for the warped AdS 3 black hole, Class. Quant. Grav. 27 (2010) 165012 [arXiv:0912.3584] [SPIRES].
K.S. Gupta, E. Harikumar, S. Sen and M. Sivakumar, The central charge of the warped AdS 3 black hole, Mod. Phys. Lett. A 25 (2010) 2065 [arXiv:1003.0991] [SPIRES].
B. Chen and Z.-b. Xu, Quasi-normal modes of warped black holes and warped AdS/CFT correspondence, JHEP 11 (2009) 091 [arXiv:0908.0057] [SPIRES].
H.-C. Kao and W.-Y. Wen, Absorption cross section in warped AdS 3 black hole revisited, JHEP 09 (2009) 102 [arXiv:0907.5555] [SPIRES].
O. Mišković and R. Olea, Background-independent charges in topologically massive gravity, JHEP 12 (2009) 046 [arXiv:0909.2275] [SPIRES].
S.K. Chakrabarti, P.R. Giri and K.S. Gupta, Scalar field dynamics in warped AdS 3 black hole background, Phys. Lett. B 680 (2009) 500 [arXiv:0903.1537] [SPIRES].
D. Anninos, M. Esole and M. Guica, Stability of warped AdS 3 vacua of topologically massive gravity, JHEP 10 (2009) 083 [arXiv:0905.2612] [SPIRES].
M. Blagojevic and B. Cvetkovic, Asymptotic structure of topologically massive gravity in spacelike stretched AdS sector, JHEP 09 (2009) 006 [arXiv:0907.0950] [SPIRES].
G. Compere and S. Detournay, Semi-classical central charge in topologically massive gravity, Class. Quant. Grav. 26 (2009) 012001 [arXiv:0808.1911] [SPIRES].
G. Compere and S. Detournay, Boundary conditions for spacelike and timelike warped AdS 3 spaces in topologically massive gravity, JHEP 08 (2009) 092 [arXiv:0906.1243] [SPIRES].
D. Anninos, W. Li, M. Padi, W. Song and A. Strominger, Warped AdS 3 black holes, JHEP 03 (2009) 130 [arXiv:0807.3040] [SPIRES].
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] [SPIRES].
N. Seiberg and E. Witten, The D1/D5 system and singular CFT, JHEP 04 (1999) 017 [hep-th/9903224] [SPIRES].
J.R. David, G. Mandal and S.R. Wadia, Microscopic formulation of black holes in string theory, Phys. Rept. 369 (2002) 549 [hep-th/0203048] [SPIRES].
A. Pakman, L. Rastelli and S.S. Razamat, A spin chain for the symmetric product CFT 2, JHEP 05 (2010) 099 [arXiv:0912.0959] [SPIRES].
G. Giribet, A. Pakman and L. Rastelli, Spectral flow in AdS 3 /CFT 2, JHEP 06 (2008) 013 [arXiv:0712.3046] [SPIRES].
A. Dabholkar and A. Pakman, Exact chiral ring of AdS 3 /CFT 2, Adv. Theor. Math. Phys. 13 (2009) 409 [hep-th/0703022] [SPIRES].
A. Pakman and A. Sever, Exact N =4 correlators of AdS 3 /CFT 2, Phys. Lett. B 652 (2007) 60 [arXiv:0704.3040] [SPIRES].
M.R. Gaberdiel and I. Kirsch, Worldsheet correlators in AdS 3 /CFT 2, JHEP 04 (2007) 050 [hep-th/0703001] [SPIRES].
S.K. Ashok, R. Benichou and J. Troost, Asymptotic symmetries of string theory on AdS 3 ×S 3 with Ramond-Ramond fluxes, JHEP 10 (2009) 051 [arXiv:0907.1242] [SPIRES].
A. Giveon, D. Kutasov and N. Seiberg, Comments on string theory on AdS 3, Adv. Theor. Math. Phys. 2 (1998) 733 [hep-th/9806194] [SPIRES].
J. de Boer, H. Ooguri, H. Robins and J. Tannenhauser, String theory on AdS 3, JHEP 12 (1998) 026 [hep-th/9812046] [SPIRES].
D. Kutasov, F. Larsen and R.G. Leigh, String theory in magnetic monopole backgrounds, Nucl. Phys. B 550 (1999) 183 [hep-th/9812027] [SPIRES].
D. Kutasov and N. Seiberg, More comments on string theory on AdS 3, JHEP 04 (1999) 008 [hep-th/9903219] [SPIRES].
R. Argurio, A. Giveon and A. Shomer, Superstrings on AdS 3 and symmetric products, JHEP 12 (2000) 003 [hep-th/0009242] [SPIRES].
A. Giveon and A. Pakman, More on superstrings in AdS 3 ×N, JHEP 03 (2003) 056 [hep-th/0302217] [SPIRES].
G. Compere, S. Detournay and M. Romo, Supersymmetric Gödel and warped black holes in string theory, Phys. Rev. D 78 (2008) 104030 [arXiv:0808.1912] [SPIRES].
D. Anninos, Hopfing and puffing warped Anti-de Sitter space, JHEP 09 (2009) 075 [arXiv:0809.2433] [SPIRES].
T.S. Levi, J. Raeymaekers, D. Van den Bleeken, W. Van Herck and B. Vercnocke, Gödel space from wrapped M2-branes, JHEP 01 (2010) 082 [arXiv:0909.4081] [SPIRES].
D. Orlando and L.I. Uruchurtu, Warped Anti-de Sitter spaces from brane intersections in type-II string theory, JHEP 06 (2010) 049 [arXiv:1003.0712] [SPIRES].
D. Israel, Quantization of heterotic strings in a Gödel/Anti de Sitter spacetime and chronology protection, JHEP 01 (2004) 042 [hep-th/0310158] [SPIRES].
D. Israel, C. Kounnas, D. Orlando and P.M. Petropoulos, Electric/magnetic deformations of S 3 and AdS 3 and geometric cosets, Fortsch. Phys. 53 (2005) 73 [hep-th/0405213] [SPIRES].
S. Detournay, D. Orlando, P.M. Petropoulos and P. Spindel, Three-dimensional black holes from deformed Anti de Sitter, JHEP 07 (2005) 072 [hep-th/0504231] [SPIRES].
K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [SPIRES].
D.T. Son, Toward an AdS/cold atoms correspondence: a geometric realization of the Schrödinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [SPIRES].
E. D’Hoker and P. Kraus, Holographic metamagnetism, quantum criticality and crossover behavior, JHEP 05 (2010) 083 [arXiv:1003.1302] [SPIRES].
J.M. Maldacena and H. Ooguri, Strings in AdS 3 and SL(2,R) WZW model. I, J. Math. Phys. 42 (2001) 2929 [hep-th/0001053] [SPIRES].
J.M. Maldacena, H. Ooguri and J. Son, Strings in AdS 3 and the SL(2,R) WZW model. II: euclidean black hole, J. Math. Phys. 42 (2001) 2961 [hep-th/0005183] [SPIRES].
J.M. Maldacena and H. Ooguri, Strings in AdS 3 and the SL(2,R) WZW model. III: correlation functions, Phys. Rev. D 65 (2002) 106006 [hep-th/0111180] [SPIRES].
S. Hemming and E. Keski-Vakkuri, The spectrum of strings on BTZ black holes and spectral flow in the SL(2,R) WZW model, Nucl. Phys. B 626 (2002) 363 [hep-th/0110252] [SPIRES].
M. Rangamani and S.F. Ross, Winding tachyons in BTZ, Phys. Rev. D 77 (2008) 026010 [arXiv:0706.0663] [SPIRES].
J. Parsons and S.F. Ross, Strings in extremal BTZ black holes, JHEP 04 (2009) 134 [arXiv:0901.3044] [SPIRES].
A.A. Tseytlin, Effective action of gauged WZW model and exact string solutions, Nucl. Phys. B 399 (1993) 601 [hep-th/9301015] [SPIRES].
A.A. Tseytlin, Conformal σ-models corresponding to gauged Wess-Zumino-Witten theories, Nucl. Phys. B 411 (1994) 509 [hep-th/9302083] [SPIRES].
S. Förste and D. Roggenkamp, Current current deformations of conformal field theories and WZW models, JHEP 05 (2003) 071 [hep-th/0304234] [SPIRES].
M. Cvetič, H. Lü and C.N. Pope, Consistent Kaluza-Klein sphere reductions, Phys. Rev. D 62 (2000) 064028 [hep-th/0003286] [SPIRES].
S.A. Apikian and C.J. Efthimiou, Minimal models of CFT on Z (N)-surfaces, Int. J. Mod. Phys. A 12 (1997) 4291 [hep-th/9610051] [SPIRES].
J.M. Maldacena and A. Strominger, AdS 3 black holes and a stringy exclusion principle, JHEP 12 (1998) 005 [hep-th/9804085] [SPIRES].
K. Hosomichi, N =2 Liouville theory with boundary, JHEP 12 (2006) 061 [hep-th/0408172] [SPIRES].
M. Bañados, G. Barnich, G. Compere and A. Gomberoff, Three dimensional origin of Gödel spacetimes and black holes, Phys. Rev. D 73 (2006) 044006 [hep-th/0512105] [SPIRES].
M. Bañados, Embeddings of the Virasoro algebra and black hole entropy, Phys. Rev. Lett. 82 (1999) 2030 [hep-th/9811162] [SPIRES].
E.J. Martinec and W. McElgin, String theory on AdS orbifolds, JHEP 04 (2002) 029 [hep-th/0106171] [SPIRES].
E.J. Martinec and W. McElgin, Exciting AdS orbifolds, JHEP 10 (2002) 050 [hep-th/0206175] [SPIRES].
G. Compere and S. Detournay, Centrally extended symmetry algebra of asymptotically Gödel spacetimes, JHEP 03 (2007) 098 [hep-th/0701039] [SPIRES].
T. Hartman and A. Strominger, Central charge for AdS 2 quantum gravity, JHEP 04 (2009) 026 [arXiv:0803.3621] [SPIRES].
T. Hartman, K. Murata, T. Nishioka and A. Strominger, CFT duals for extreme black holes, JHEP 04 (2009) 019 [arXiv:0811.4393] [SPIRES].
M. Guica, K. Skenderis, M. Taylor and B.C. van Rees, Holography for Schrödinger backgrounds, arXiv:1008.1991 [SPIRES].
V.A. Fateev and A.B. Zamolodchikov, Parafermionic currents in the two-dimensional conformal quantum field theory and selfdual critical points in Z (n) invariant statistical systems, Sov. Phys. JETP 62 (1985) 215 [SPIRES].
J.D. Lykken, Finitely reducible realizations of the N =2 superconformal algebra, Nucl. Phys. B 313 (1989) 473 [SPIRES].
A. Gerasimov, A. Marshakov and A. Morozov, Free field representation of parafermions and related coset models, Nucl. Phys. B 328 (1989) 664 [SPIRES].
D. Karabali and H.J. Schnitzer, BRST quantization of the gauged WZW action and coset conformal field theories, Nucl. Phys. B 329 (1990) 649 [SPIRES].
S. Hwang and H. Rhedin, The BRST formulation of G/H WZNW models, Nucl. Phys. B 406 (1993) 165 [hep-th/9305174] [SPIRES].
G. Barnich and F. Brandt, Covariant theory of asymptotic symmetries, conservation laws and central charges, Nucl. Phys. B 633 (2002) 3 [hep-th/0111246] [SPIRES].
G. Barnich, Boundary charges in gauge theories: using Stokes theorem in the bulk, Class. Quant. Grav. 20 (2003) 3685 [hep-th/0301039] [SPIRES].
G. Barnich and G. Compere, Surface charge algebra in gauge theories and thermodynamic integrability, J. Math. Phys. 49 (2008) 042901 [arXiv:0708.2378] [SPIRES].
G. Compere, Symmetries and conservation laws in Lagrangian gauge theories with applications to the mechanics of black holes and to gravity in three dimensions, arXiv:0708.3153 [SPIRES].
G. Compere, S. de Buyl, S. Detournay and K. Yoshida, Asymptotic symmetries of Schrödinger spacetimes, JHEP 10 (2009) 032 [arXiv:0908.1402] [SPIRES].
G. Barnich, F. Brandt and M. Henneaux, Local BRST cohomology in the antifield formalism. 1. General theorems, Commun. Math. Phys. 174 (1995) 57 [hep-th/9405109] [SPIRES].
G. Barnich and G. Compere, Generalized Smarr relation for Kerr AdS black holes from improved surface integrals, Phys. Rev. D 71 (2005) 044016 [gr-qc/0412029] [SPIRES].
G. Barnich and G. Compere, Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions, Class. Quant. Grav. 24 (2007) F15 [gr-qc/0610130] [SPIRES].
L.F. Abbott and S. Deser, Stability of gravity with a cosmological constant, Nucl. Phys. B 195 (1982) 76 [SPIRES].
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] [SPIRES].
G. Compere, Note on the first law with p-form potentials, Phys. Rev. D 75 (2007) 124020 [hep-th/0703004] [SPIRES].
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Detournay, S., Israël, D., Lapan, J.M. et al. String theory on warped AdS3 and Virasoro resonances. J. High Energ. Phys. 2011, 30 (2011). https://doi.org/10.1007/JHEP01(2011)030
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DOI: https://doi.org/10.1007/JHEP01(2011)030