Abstract
It has been recently argued that the averaging of free CFT’s over the Narain lattice can be holographically described through a Chern-Simons theory for U (1)D ×U (1)D with a precise prescription to sum over three-dimensional handlebodies.
We show that a gravitational dual of these averaged CFT’s would be provided by Einstein gravity on AdS3 with U (1)D−1 × U (1)D−1 gauge fields, endowed with a precise set of boundary conditions closely related to the “soft hairy” ones. Gravitational excitations then go along diagonal SL (2, ℝ) generators, so that the asymptotic symmetries are spanned by U (1)D × U (1)D currents. The stress-energy tensor can then be geometrically seen as composite of these currents through a twisted Sugawara construction. Our boundary conditions are such that for the reduced phase space, there is a one-to-one map between the configurations in the gravitational and the purely abelian theories. The partition function in the bulk could then also be performed either from a non-abelian Chern-Simons theory for two copies of SL (2, ℝ) × U (1)D−1 generators, or formally through a path integral along the family of allowed configurations for the metric. The new boundary conditions naturally accommodate BTZ black holes, and the microscopic number of states then appears to be manifestly positive and suitably accounted for from the partition function in the bulk. The inclusion of higher spin currents through an extended twisted Sugawara construction in the context of higher spin gravity is also briefly addressed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Teitelboim, Gravitation and Hamiltonian Structure in Two Space-Time Dimensions, Phys. Lett. B 126 (1983) 41 [INSPIRE].
R. Jackiw, Liouville field theory: A two-dimensional model for gravity?, in Quantum Theory Of Gravity, S. Christensen ed., Adam Hilger, Bristol, pp. 403–420 (1984) [INSPIRE].
C. Teitelboim, The Hamiltonian structure of two-dimensional space-time and its relation with the conformal anomaly, in Quantum Theory Of Gravity, S. Christensen ed., Adam Hilger, Bristol, pp. 327–344 (1984) [INSPIRE].
R. Jackiw, Lower Dimensional Gravity, Nucl. Phys. B 252 (1985) 343 [INSPIRE].
P. Saad, S.H. Shenker and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115 [INSPIRE].
A. Maloney and E. Witten, Averaging Over Narain Moduli Space, arXiv:2006.04855 [INSPIRE].
N. Afkhami-Jeddi, H. Cohn, T. Hartman and A. Tajdini, Free partition functions and an averaged holographic duality, arXiv:2006.04839 [INSPIRE].
K.S. Narain, New Heterotic String Theories in Uncompactified Dimensions < 10, Phys. Lett. B 169 (1986) 41 [INSPIRE].
K.S. Narain, M.H. Sarmadi and E. Witten, A Note on Toroidal Compactification of Heterotic String Theory, Nucl. Phys. B 279 (1987) 369 [INSPIRE].
C.L. Siegel, Indefinite quadratische formen und funktionentheorie I, Math. Ann. 124 (1951) 17.
H. Maass and T. Srinivasan, Lectures on Siegels modular functions, Tata Institute of Fundamental Research (1955).
A. Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964) 143.
A. Weil, Sur la formule de siegel dans la théorie des groupes classiques, Acta Math. 113 (1965) 1.
M. Porrati and C. Yu, Kac-Moody and Virasoro Characters from the Perturbative Chern-Simons Path Integral, JHEP 05 (2019) 083 [arXiv:1903.05100] [INSPIRE].
A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three Dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE].
H. Afshar et al., Soft Heisenberg hair on black holes in three dimensions, Phys. Rev. D 93 (2016) 101503 [arXiv:1603.04824] [INSPIRE].
H. Afshar, D. Grumiller, W. Merbis, A. Perez, D. Tempo and R. Troncoso, Soft hairy horizons in three spacetime dimensions, Phys. Rev. D 95 (2017) 106005 [arXiv:1611.09783] [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].
O. Coussaert, M. Henneaux and P. van Driel, The Asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant, Class. Quant. Grav. 12 (1995) 2961 [gr-qc/9506019] [INSPIRE].
M. Henneaux, A. Perez, D. Tempo and R. Troncoso, Chemical potentials in three-dimensional higher spin anti-de Sitter gravity, JHEP 12 (2013) 048 [arXiv:1309.4362] [INSPIRE].
C. Bunster, M. Henneaux, A. Perez, D. Tempo and R. Troncoso, Generalized Black Holes in Three-dimensional Spacetime, JHEP 05 (2014) 031 [arXiv:1404.3305] [INSPIRE].
T. Regge and C. Teitelboim, Role of Surface Integrals in the Hamiltonian Formulation of General Relativity, Annals Phys. 88 (1974) 286 [INSPIRE].
G. Barnich and F. Brandt, Covariant theory of asymptotic symmetries, conservation laws and central charges, Nucl. Phys. B 633 (2002) 3 [hep-th/0111246] [INSPIRE].
A. Pérez, D. Tempo and R. Troncoso, Boundary conditions for General Relativity on AdS3 and the KdV hierarchy, JHEP 06 (2016) 103 [arXiv:1605.04490] [INSPIRE].
O. Fuentealba et al., Integrable systems with BMS3 Poisson structure and the dynamics of locally flat spacetimes, JHEP 01 (2018) 148 [arXiv:1711.02646] [INSPIRE].
D. Melnikov, F. Novaes, A. Pérez and R. Troncoso, Lifshitz Scaling, Microstate Counting from Number Theory and Black Hole Entropy, JHEP 06 (2019) 054 [arXiv:1808.04034] [INSPIRE].
E. Ojeda and A. Pérez, Boundary conditions for General Relativity in three-dimensional spacetimes, integrable systems and the KdV/mKdV hierarchies, JHEP 08 (2019) 079 [arXiv:1906.11226] [INSPIRE].
D. Grumiller and W. Merbis, Near horizon dynamics of three dimensional black holes, SciPost Phys. 8 (2020) 010 [arXiv:1906.10694] [INSPIRE].
J. 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. Henneaux, L. Maoz and A. Schwimmer, Asymptotic dynamics and asymptotic symmetries of three-dimensional extended AdS supergravity, Annals Phys. 282 (2000) 31 [hep-th/9910013] [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. 88 (2013) 069902] [gr-qc/9302012] [INSPIRE].
D. Grumiller, A. Perez, S. Prohazka, D. Tempo and R. Troncoso, Higher Spin Black Holes with Soft Hair, JHEP 10 (2016) 119 [arXiv:1607.05360] [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].
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].
M.A. Vasiliev, Higher spin gauge theories in four-dimensions, three-dimensions, and two-dimensions, Int. J. Mod. Phys. D 5 (1996) 763 [hep-th/9611024] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2006.08216
Rights and permissions
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.
About this article
Cite this article
Pérez, A., Troncoso, R. Gravitational dual of averaged free CFT’s over the Narain lattice. J. High Energ. Phys. 2020, 15 (2020). https://doi.org/10.1007/JHEP11(2020)015
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2020)015