A two-dimensional CFT dual to a semiclassical theory of gravity in three dimensions must have a large central charge c and a sparse low energy spectrum. This constrains the OPE coefficients and density of states of the CFT via the conformal bootstrap. We define an ensemble of CFT data by averaging over OPE coefficients subject to these bootstrap constraints, and show that calculations in this ensemble reproduce semiclassical 3D gravity. We analyze a wide variety of gravitational solutions, both in pure Einstein gravity and gravity coupled to massive point particles, including Euclidean wormholes with multiple boundaries and higher topology spacetimes with a single boundary. In all cases we find that the on-shell action of gravity agrees with the ensemble-averaged CFT at large c. The one-loop corrections also match in the cases where they have been computed. We also show that the bulk effective theory has random couplings induced by wormholes, providing a controlled, semiclassical realization of the mechanism of Coleman, Giddings, and Strominger.
J.L. Cardy, Operator content of two-dimensional conformally invariant theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].
J.M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43 (1991) 2046.
A. Belin, J. De Boer, P. Nayak and J. Sonner, Charged eigenstate thermalization, Euclidean wormholes and global symmetries in quantum gravity, SciPost Phys. 12 (2022) 059 [arXiv:2012.07875] [INSPIRE].
S.R. Coleman, Black holes as red herrings: topological fluctuations and the loss of quantum coherence, Nucl. Phys. B 307 (1988) 867 [INSPIRE].
S.B. Giddings and A. Strominger, Axion induced topology change in quantum gravity and string theory, Nucl. Phys. B 306 (1988) 890 [INSPIRE].
S.B. Giddings and A. Strominger, Loss of incoherence and determination of coupling constants in quantum gravity, Nucl. Phys. B 307 (1988) 854 [INSPIRE].
E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
S. Elitzur, G.W. Moore, A. Schwimmer and N. Seiberg, Remarks on the canonical quantization of the Chern-Simons-Witten theory, Nucl. Phys. B 326 (1989) 108 [INSPIRE].
H.L. Verlinde, Conformal field theory, 2D quantum gravity and quantization of Teichmüller space, Nucl. Phys. B 337 (1990) 652 [INSPIRE].
P.G. Zograf and L.A. Takhtadzhyan, On the Liouville equation, accessory parameters and the geometry of Teichmüller space for Riemann surfaces of genus 0, Math. U.S.S.R.-Sb. 60 (1988) 143.
P.G. Zograf and L.A. Takhtadzhyan, On the uniformization of Riemann surfaces and on the Weil-Petersson metric on the Teichmüller and Schottky spaces, Math. U.S.S.R.-Sb. 60 (1988) 297.
L. Takhtajan and P. Zograf, Hyperbolic 2-spheres with conical singularities, accessory parameters and Kähler metrics on M0,n, Trans. Amer. Math. Soc. 355 (2002) 1857.
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.CV/0204318] [INSPIRE].
K.K. Uhlenbeck, Closed minimal surfaces in hyperbolic 3-manifolds, in Seminar on minimal submanifolds, Ann. Math. Stud. 103 (1983) 147.
C.T. McMullen, The moduli space of Riemann surfaces is Kähler hyperbolic, Ann. Math. 151 (2000) 327.
P.G. Zograf and L.A. Takhtadzhyan, Action of the Liouville equation is a generating function for the accessory parameters and the potential of the Weil-Petersson metric on the Teichmüller space, Funct. Anal. Appl. 19 (1986) 219 [Funkt. Anal. Pril. 19 (1985) 67].
L. Takhtajan, Semi-classical Liouville theory, complex geometry of moduli spaces, and uniformization of Riemann surfaces, in New symmetry principles in quantum field theory (Cargèse, 1991), NATO Adv. Sci. Inst. B 295, Plenum, New York, U.S.A. (1992), p. 383.
J. Park, L.A. Takhtajan and L.-P. Teo, Potentials and Chern forms for Weil-Petersson and Takhtajan-Zograf metrics on moduli spaces, Adv. Math. 305 (2017) 856.
S.B. Giddings and A. Strominger, Baby universes, third quantization and the cosmological constant, Nucl. Phys. B 321 (1989) 481 [INSPIRE].
S.B. Giddings and A. Strominger, String wormholes, Phys. Lett. B 230 (1989) 46 [INSPIRE].
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
A.B. Zamolodchikov, Conformal symmetry in two-dimensional space: recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088.
T. Banks, Prolegomena to a theory of bifurcating universes: a nonlocal solution to the cosmological constant problem or little lambda goes back to the future, Nucl. Phys. B 309 (1988) 493 [INSPIRE].
L.A. Takhtajan and P. Zograf, Local index theorem for orbifold Riemann surfaces, Lett. Math. Phys. 109 (2018) 1119.
E. D’Hoker and D.H. Phong, On determinants of Laplacians on Riemann surfaces, Commun. Math. Phys. 104 (1986) 537 [INSPIRE].
P. Sarnak, Determinants of Laplacians, Commun. Math. Phys. 110 (1987) 113.
A. Mcintyre and L.-P. Teo, Holomorphic factorization of determinants of Laplacians using quasi-Fuchsian uniformization, Lett. Math. Phys. 83 (2007) 41 [math.CV/0605605].
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Chandra, J., Collier, S., Hartman, T. et al. Semiclassical 3D gravity as an average of large-c CFTs. J. High Energ. Phys. 2022, 69 (2022). https://doi.org/10.1007/JHEP12(2022)069
- AdS-CFT Correspondence
- Field Theories in Lower Dimensions
- Black Holes
- Black Holes in String Theory