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Quantum gravity partition functions in three dimensions. (English) Zbl 1270.83022
Summary: We consider pure three-dimensional quantum gravity with a negative cosmological constant. The sum of known contributions to the partition function from classical geometries can be computed exactly, including quantum corrections. However, the result is not physically sensible, and if the model does exist, there are some additional contributions. One possibility is that the theory may have long strings and a continuous spectrum. Another possibility is that complex geometries need to be included, possibly leading to a holomorphically factorized partition function. We analyze the subleading corrections to the Bekenstein-Hawking entropy and show that these can be correctly reproduced in such a holomorphically factorized theory. We also consider the Hawking-Page phase transition between a thermal gas and a black hole and show that it is a phase transition of Lee-Yang type, associated with a condensation of zeros in the complex temperature plane. Finally, we analyze pure three-dimensional supergravity, with similar results.

##### MSC:
 83C45 Quantization of the gravitational field 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 83C57 Black holes 94A17 Measures of information, entropy 83E50 Supergravity
##### Keywords:
 [1] Deser, S.; Jackiw, R., Three-dimensional cosmological gravity: dynamics of constant curvature, Annals Phys., 153, 405, (1984) [2] Brown, JD; Henneaux, M., Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys., 104, 207, (1986) [3] Bañados, M.; Teitelboim, C.; Zanelli, J., The black hole in three-dimensional space-time, Phys. Rev. Lett., 69, 1849, (1992) [4] Bañados, M.; Henneaux, M.; Teitelboim, C.; Zanelli, J., Geometry of the (2 + 1) black hole, Phys. Rev., D 48, 1506, (1993) [5] Maldacena, JM, The large-$$N$$ limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys., 2, 231, (1998) [6] Strominger, A., Black hole entropy from near-horizon microstates, JHEP, 02, 009, (1998) [7] Carlip, S., Conformal field theory, (2 + 1)-dimensional gravity and the BTZ black hole, Class. Quant. Grav., 22, 85, (2005) [8] Gibbons, GW; Hawking, SW; Perry, MJ, Path integrals and the indefiniteness of the gravitational action, Nucl. Phys., B 138, 141, (1978) [9] Maldacena, JM; Strominger, A., Ads_{3} black holes and a stringy exclusion principle, JHEP, 12, 005, (1998) [10] R. Dijkgraaf, J.M. Maldacena, G.W. Moore and E.P. Verlinde, A black hole Farey tail, hep-th/0005003 [SPIRES]. [11] Ray, DB; Singer, IM, Analytic torsion for complex manifolds, Annals Math., 98, 154, (1973) [12] X. Yin, Partition functions of three-dimensional pure gravity, arXiv:0710.2129 [SPIRES]. [13] E. Witten, Three-dimensional gravity revisited, arXiv:0706.3359 [SPIRES]. [14] Manschot, J., Ads_{3} partition functions reconstructed, JHEP, 10, 103, (2007) [15] Hawking, SW; Page, DN, Thermodynamics of black holes in anti-de Sitter space, Commun. Math. Phys., 87, 577, (1983) [16] Carlip, S.; Teitelboim, C., Aspects of black hole quantum mechanics and thermodynamics in (2 + 1)-dimensions, Phys. Rev., D 51, 622, (1995) [17] Deser, S.; Jackiw, R.; ’t Hooft, G., Three-dimensional Einstein gravity: dynamics of flat space, Ann. Phys., 152, 220, (1984) [18] Achucarro, A.; Townsend, PK, A Chern-Simons action for three-dimensional anti-de Sitter supergravity theories, Phys. Lett., B 180, 89, (1986) [19] Witten, E., (2 + 1)-dimensional gravity as an exactly soluble system, Nucl. Phys., B 311, 46, (1988) [20] Witten, E., Coadjoint orbits of the Virasoro group, Commun. Math. Phys., 114, 1, (1988) [21] Bytsenko, AA; Vanzo, L.; Zerbini, S., Quantum correction to the entropy of the (2+1)-dimensional black hole, Phys. Rev., D 57, 4917, (1998) [22] H. Iwaniec, Spectral methods of automorphic forms, American Mathematical Society, U.S.A. (2002). [23] Kleban, M.; Porrati, M.; Rabadán, R., Poincaré recurrences and topological diversity, JHEP, 10, 030, (2004) [24] Maldacena, JM; Michelson, J.; Strominger, A., Anti-de Sitter fragmentation, JHEP, 02, 011, (1999) [25] Seiberg, N.; Witten, E., The D1/D5 system and singular CFT, JHEP, 04, 017, (1999) [26] Krasnov, K., On holomorphic factorization in asymptotically AdS 3D gravity, Class. Quant. Grav., 20, 4015, (2003) [27] G. Höhn, Selbstduale Vertexoperatorsuperalgebrenund das Babymonster, Ph.D. thesis, University of Bonn, Bonn ,Germany 1995), Bonner Mathematische Schriften volume 286 (1996) 1, English translation, arXiv:0706.0236. [28] Gaiotto, D.; Yin, X., Genus two partition functions of extremal conformal field theories, JHEP, 08, 029, (2007) [29] X. Yin, Partition functions of three-dimensional pure gravity, arXiv:0710.2129 [SPIRES]. [30] Gaberdiel, MR, Constraints on extremal self-dual cfts, JHEP, 11, 087, (2007) [31] D. Gaiotto, to appear. [32] I.B. Frenkel, J. Lepowsky and A. Meurman, Vertex operator algebras and the monster, Academic Press, Boston, Boston U.S.A. (1988). [33] Carlip, S., Logarithmic corrections to black hole entropy from the cardy formula, Class. Quant. Grav., 17, 4175, (2000) [34] Yang, CN; Lee, T-D, Statistical theory of equations of state and phase transitions. I. theory of condensation, Phys. Rev., 87, 404, (1952) [35] Fisher, M., The nature of critical points, (1965), Boulder U.S.A. [36] Rankin, R., The zeros of certain Poincaré series, Compos. Math., 46, 255, (1982) [37] Asai, T.; Kaneko, M.; Ninomiya, H., Zeros of certain modular functions and an application, Commentarii Math. Univ. Sancti Pauli, 46, 93, (1997) [38] Bekenstein, JD, A universal upper bound on the entropy to energy ratio for bounded systems, Phys. Rev., D 23, 287, (1981) [39] T. Apostol, Modular functions and Dirichlet series in number theory, Springer Verlag, Germany (1990). [40] M. Kaneko, private communication.