Abstract
The 1-loop partition function of the handlebody solutions in the AdS3 gravity have been derived some years ago using the heat kernel techniques and the method of images. In the semiclassical limit, such partition function should correspond to the order O(c 0) part in the partition function of dual conformal field theory(CFT) on the boundary Riemann surface. The higher genus partition function could be computed by the multi-point functions in the Riemann sphere via sewing prescription. In the large central charge limit, the CFT is effectively free in the sense that to the leading order of c the multi-point function is further simplified to be a summation over the products of two-point functions of single-particle states. Correspondingly in the bulk, the graviton is freely propagating without interaction. Furthermore the product of the two-point functions may define the links, each of which is in one-to-one correspondence with the conjugacy class of the Schottky group of the Riemann surface. Moreover, the value of a link is determined by the multiplier of the element in the conjugacy class. This allows us to reproduce exactly the gravitational 1-loop partition function. The proof can be generalized to the higher spin gravity and its dual CFT.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
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].
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].
M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, Geometry of the (2 + 1) black hole, Phys. Rev. D 48 (1993) 1506 [Erratum ibid. D 88 (2013) 069902] [gr-qc/9302012] [INSPIRE].
A. Strominger, Black hole entropy from near horizon microstates, JHEP 02 (1998) 009 [hep-th/9712251] [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].
E. Witten, Three-dimensional gravity revisited, arXiv:0706.3359 [INSPIRE].
M.P. Blencowe, A consistent interacting massless higher spin field theory in D = (2 + 1), Class. Quant. Grav. 6 (1989) 443 [INSPIRE].
E. Bergshoeff, M.P. Blencowe and K.S. Stelle, Area preserving diffeomorphisms and higher spin algebra, Commun. Math. Phys. 128 (1990) 213 [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].
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].
K. Krasnov, Holography and Riemann surfaces, Adv. Theor. Math. Phys. 4 (2000) 929 [hep-th/0005106] [INSPIRE].
P.G. Zograf and L.A. Takhtajan, On uniformization of Riemann surfaces and the Weil-Petersson metric on Teichmüller and Schottky spaces, Math. USSR. Sb. 60 (1988) 297.
L.A. Takhtajan and L.-P. Teo, Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography, Commun. Math. Phys. 239 (2003) 183 [math/0204318] [INSPIRE].
X. Yin, On non-handlebody instantons in 3D gravity, JHEP 09 (2008) 120 [arXiv:0711.2803] [INSPIRE].
X. Yin, Partition functions of three-dimensional pure gravity, Commun. Num. Theor. Phys. 2 (2008) 285 [arXiv:0710.2129] [INSPIRE].
S. Giombi, A. Maloney and X. Yin, One-loop partition functions of 3D gravity, JHEP 08 (2008) 007 [arXiv:0804.1773] [INSPIRE].
T. Hartman, Entanglement entropy at large central charge, arXiv:1303.6955 [INSPIRE].
T. Hartman, C.A. Keller and B. Stoica, Universal spectrum of 2d conformal field theory in the large c limit, JHEP 09 (2014) 118 [arXiv:1405.5137] [INSPIRE].
A. Maloney and E. Witten, Quantum gravity partition functions in three dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE].
T. Faulkner, The entanglement Rényi entropies of disjoint intervals in AdS/CFT, arXiv:1303.7221 [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
M. Headrick, Entanglement Rényi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].
T. Barrella, X. Dong, S.A. Hartnoll and V.L. Martin, Holographic entanglement beyond classical gravity, JHEP 09 (2013) 109 [arXiv:1306.4682] [INSPIRE].
B. Chen and J.-J. Zhang, On short interval expansion of Rényi entropy, JHEP 11 (2013) 164 [arXiv:1309.5453] [INSPIRE].
B. Chen, J. Long and J.-J. Zhang, Holographic Rényi entropy for CFT with W symmetry, JHEP 04 (2014) 041 [arXiv:1312.5510] [INSPIRE].
E. Perlmutter, Comments on Rényi entropy in AdS 3 /CFT 2, JHEP 05 (2014) 052 [arXiv:1312.5740] [INSPIRE].
B. Chen, F.-Y. Song and J.-J. Zhang, Holographic Rényi entropy in AdS 3 /LCFT 2 correspondence, JHEP 03 (2014) 137 [arXiv:1401.0261] [INSPIRE].
M. Beccaria and G. Macorini, On the next-to-leading holographic entanglement entropy in AdS 3 /CFT 2, JHEP 04 (2014) 045 [arXiv:1402.0659] [INSPIRE].
B. Chen and J.-Q. Wu, Single interval Rényi entropy at low temperature, JHEP 08 (2014) 032 [arXiv:1405.6254] [INSPIRE].
B. Chen and J.-Q. Wu, Holographic calculation for large interval Rényi entropy at high temperature, Phys. Rev. D 92 (2015) 106001 [arXiv:1506.03206] [INSPIRE].
B. Chen, J.-Q. Wu and Z.-C. Zheng, Holographic Rényi entropy of single interval on torus: with W symmetry, Phys. Rev. D 92 (2015) 066002 [arXiv:1507.00183] [INSPIRE].
G. Segal, The definition of conformal field theory, in Topology, geometry and quantum field theory, London Math. Soc. Lecture Note Ser. 38, Cambridge University Press, Cambridge U.K. (2004), pg. 421.
M.R. Gaberdiel, C.A. Keller and R. Volpato, Genus two partition functions of chiral conformal field theories, Commun. Num. Theor. Phys. 4 (2010) 295 [arXiv:1002.3371] [INSPIRE].
M. Headrick, A. Maloney, E. Perlmutter and I.G. Zadeh, Rényi entropies, the analytic bootstrap and 3D quantum gravity at higher genus, JHEP 07 (2015) 059 [arXiv:1503.07111] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Universality of long-distance AdS physics from the CFT bootstrap, JHEP 08 (2014) 145 [arXiv:1403.6829] [INSPIRE].
P. Caputa, M. Nozaki and T. Takayanagi, Entanglement of local operators in large-N conformal field theories, Prog. Theor. Exp. Phys. 2014 (2014) 093B06 [arXiv:1405.5946] [INSPIRE].
W. Li, W. Song and A. Strominger, Chiral gravity in three dimensions, JHEP 04 (2008) 082 [arXiv:0801.4566] [INSPIRE].
S. Deser, R. Jackiw and S. Templeton, Topologically massive gauge theories, Annals Phys. 140 (1982) 372 [Erratum ibid. 185 (1988) 406] [Erratum ibid. 281 (2000) 409] [INSPIRE].
A. Maloney, W. Song and A. Strominger, Chiral gravity, log gravity and extremal CFT, Phys. Rev. D 81 (2010) 064007 [arXiv:0903.4573] [INSPIRE].
M.R. Gaberdiel, D. Grumiller and D. Vassilevich, Graviton 1-loop partition function for 3-dimensional massive gravity, JHEP 11 (2010) 094 [arXiv:1007.5189] [INSPIRE].
B. Chen, J. Long and J.-B. Wu, Spin-3 topologically massive gravity, Phys. Lett. B 705 (2011) 513 [arXiv:1106.5141] [INSPIRE].
A. Bagchi, S. Lal, A. Saha and B. Sahoo, Topologically massive higher spin gravity, JHEP 10 (2011) 150 [arXiv:1107.0915] [INSPIRE].
A. Bagchi, S. Lal, A. Saha and B. Sahoo, One loop partition function for topologically massive higher spin gravity, JHEP 12 (2011) 068 [arXiv:1107.2063] [INSPIRE].
B. Chen and J. Long, High spin topologically massive gravity, JHEP 12 (2011) 114 [arXiv:1110.5113] [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: 1509.02062
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
Chen, B., Wu, Jq. 1-loop partition function in AdS 3/CFT 2 . J. High Energ. Phys. 2015, 1–29 (2015). https://doi.org/10.1007/JHEP12(2015)109
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/JHEP12(2015)109