Abstract
We discuss the quantization and holographic aspects of a class of conical spaces in 2+1 dimensional pure AdS gravity. These appear as topological solitons in the Chern-Simons formulation of the theory and are closely related to the recently studied conical solutions in higher spin gravity. We discuss the classical fluctuations around these solutions, which form exceptional coadjoint orbits of the asymptotic Virasoro group. We argue that the quantization of these solutions leads to nonunitary representations of the Virasoro algebra, on account of their having boundary graviton fluctuations which lower the energy. We propose a framework to quantize them in a semiclassical expansion in the inverse central charge, which we use to compute their one-loop corrected energies. Interestingly, the resulting Virasoro representations contain a null vector, thus providing an appearance of Kac’s degenerate representations, which are nonunitary at large central charge, in the context of gravity. We match the computed quantum corrections in the bulk with the properties of a class of primaries in Kac’s classification.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
M. Bañados, C. Teitelboim and J. Zanelli, The black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].
A. Strominger, Black hole entropy from near horizon microstates, JHEP 02 (1998) 009 [hep-th/9712251] [INSPIRE].
E. Witten, Three-Dimensional Gravity Revisited, arXiv:0706.3359 [INSPIRE].
M.R. Gaberdiel, Constraints on extremal self-dual CFTs, JHEP 11 (2007) 087 [arXiv:0707.4073] [INSPIRE].
A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three Dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE].
A. Achucarro and P.K. Townsend, A Chern-Simons Action for Three-Dimensional anti-de Sitter Supergravity Theories, Phys. Lett. B 180 (1986) 89 [INSPIRE].
E. Witten, (2+1)-Dimensional Gravity as an Exactly Soluble System, Nucl. Phys. B 311 (1988) 46 [INSPIRE].
S. Deser, R. Jackiw and G. ’t Hooft, Three-Dimensional Einstein Gravity: Dynamics of Flat Space, Annals Phys. 152 (1984) 220 [INSPIRE].
S. Deser and R. Jackiw, Three-Dimensional Cosmological Gravity: Dynamics of Constant Curvature, Annals Phys. 153 (1984) 405 [INSPIRE].
J.M. Izquierdo and P.K. Townsend, Supersymmetric space-times in (2+1) AdS supergravity models, Class. Quant. Grav. 12 (1995) 895 [gr-qc/9501018] [INSPIRE].
T. Mansson and B. Sundborg, Multi-black-hole sectors of AdS 3 gravity, Phys. Rev. D 65 (2002) 024025 [hep-th/0010083] [INSPIRE].
V. Balasubramanian, J. de Boer, E. Keski-Vakkuri and S.F. Ross, Supersymmetric conical defects: Towards a string theoretic description of black hole formation, Phys. Rev. D 64 (2001) 064011 [hep-th/0011217] [INSPIRE].
E. Witten, Topology Changing Amplitudes in (2+1)-Dimensional Gravity, Nucl. Phys. B 323 (1989) 113 [INSPIRE].
M. Ammon, A. Castro and N. Iqbal, Wilson Lines and Entanglement Entropy in Higher Spin Gravity, JHEP 10 (2013) 110 [arXiv:1306.4338] [INSPIRE].
D. Grumiller, W. Riedler, J. Rosseel and T. Zojer, Holographic applications of logarithmic conformal field theories, J. Phys. A 46 (2013) 494002 [arXiv:1302.0280] [INSPIRE].
C. Vafa, Non-unitary holography, arXiv:1409.1603 [INSPIRE].
A. Castro, R. Gopakumar, M. Gutperle and J. Raeymaekers, Conical Defects in Higher Spin Theories, JHEP 02 (2012) 096 [arXiv:1111.3381] [INSPIRE].
E. Perlmutter, T. Prochazka and J. Raeymaekers, The semiclassical limit of W N CFTs and Vasiliev theory, JHEP 05 (2013) 007 [arXiv:1210.8452] [INSPIRE].
E. Hijano, P. Kraus and E. Perlmutter, Matching four-point functions in higher spin AdS 3 /CF T 2, JHEP 05 (2013) 163 [arXiv:1302.6113] [INSPIRE].
A. Campoleoni and S. Fredenhagen, On the higher-spin charges of conical defects, Phys. Lett. B 726 (2013) 387 [arXiv:1307.3745] [INSPIRE].
S. Coleman, Classical lumps and their quantum descendants, in Aspects of Symmetry, Cambridge University Press (1985).
V.F. Lazutkin,T.F. Pankratova, Normal forms and versal deformations for Hill’s equation, Funkts. Anal. Prilozh. 9 (1975) 41.
G. Segal, Unitarity Representations of Some Infinite Dimensional Groups, Commun. Math. Phys. 80 (1981) 301 [INSPIRE].
E. Witten, Coadjoint Orbits of the Virasoro Group, Commun. Math. Phys. 114 (1988) 1 [INSPIRE].
A. Alekseev and S.L. Shatashvili, Path Integral Quantization of the Coadjoint Orbits of the Virasoro Group and 2D Gravity, Nucl. Phys. B 323 (1989) 719 [INSPIRE].
M. Bershadsky and H. Ooguri, Hidden SL(n) Symmetry in Conformal Field Theories, Commun. Math. Phys. 126 (1989) 49 [INSPIRE].
J. Balog, L. Feher and L. Palla, Coadjoint orbits of the Virasoro algebra and the global Liouville equation, Int. J. Mod. Phys. A 13 (1998) 315 [hep-th/9703045] [INSPIRE].
A. Garbarz and M. Leston, Classification of Boundary Gravitons in AdS 3 Gravity, JHEP 05 (2014) 141 [arXiv:1403.3367] [INSPIRE].
G. Barnich and B. Oblak, Holographic positive energy theorems in three-dimensional gravity, Class. Quant. Grav. 31 (2014) 152001 [arXiv:1403.3835] [INSPIRE].
V.G. Kac, Contravariant Form for Infinite Dimensional Lie Algebras and Superalgebras, in Austin 1978, Proceedings, Group Theoretical Methods In Physics, Berlin (1979), pg. 441.
E. Witten, Quantization of Chern-Simons Gauge Theory With Complex Gauge Group, Commun. Math. Phys. 137 (1991) 29 [INSPIRE].
M. Bañados, Three-dimensional quantum geometry and black holes, AIP Conf. Proc. 484 (1999) 147 [hep-th/9901148] [INSPIRE].
J. de Boer and J.I. Jottar, Entanglement Entropy and Higher Spin Holography in AdS 3, JHEP 04 (2014) 089 [arXiv:1306.4347] [INSPIRE].
A. Castro and E. Llabrés, Unravelling Holographic Entanglement Entropy in Higher Spin Theories, arXiv:1410.2870 [INSPIRE].
A. Castro, S. Detournay, N. Iqbal and E. Perlmutter, Holographic entanglement entropy and gravitational anomalies, JHEP 07 (2014) 114 [arXiv:1405.2792] [INSPIRE].
A.A. Kirillov, Lectures on the Orbit Method, Graduate Studies in Mathematics, Volume 64, American Mathematical Society (2004).
B.L. Feigin and D.B. Fuks, Invariant skew symmetric differential operators on the line and verma modules over the Virasoro algebra, Funct. Anal. Appl. 16 (1982) 114 [INSPIRE].
L. Benoit and Y. Saint-Aubin, Degenerate Conformal Field Theories and Explicit Expression for Some Null Vectors, Phys. Lett. B 215 (1988) 517 [INSPIRE].
H. Kodama, Holomorphic Wave Function of the Universe, Phys. Rev. D 42 (1990) 2548 [INSPIRE].
E. Witten, A Note on the Chern-Simons and Kodama wave functions, gr-qc/0306083 [INSPIRE].
M. Bauer, P. Di Francesco, C. Itzykson and J.B. Zuber, Covariant differential equations and singular vectors in Virasoro representations, Nucl. Phys. B 362 (1991) 515 [INSPIRE].
S.F. Prokushkin and M.A. Vasiliev, Higher spin gauge interactions for massive matter fields in 3D AdS space-time, Nucl. Phys. B 545 (1999) 385 [hep-th/9806236] [INSPIRE].
A. Castro, M.R. Gaberdiel, T. Hartman, A. Maloney and R. Volpato, The Gravity Dual of the Ising Model, Phys. Rev. D 85 (2012) 024032 [arXiv:1111.1987] [INSPIRE].
M. Henneaux and S.-J. Rey, Nonlinear W ∞ as Asymptotic Symmetry of Three-Dimensional Higher Spin Anti-de Sitter Gravity, JHEP 12 (2010) 007 [arXiv:1008.4579] [INSPIRE].
A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields, JHEP 11 (2010) 007 [arXiv:1008.4744] [INSPIRE].
A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, Towards metric-like higher-spin gauge theories in three dimensions, J. Phys. A 46 (2013) 214017 [arXiv:1208.1851] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1412.0278
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Raeymaekers, J. Quantization of conical spaces in 3D gravity. J. High Energ. Phys. 2015, 60 (2015). https://doi.org/10.1007/JHEP03(2015)060
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2015)060