Abstract
Any unitary compact two-dimensional CFT with c > 1 and no symmetries beyond Virasoro has a parametrically large density of primary states at large spin for \( \overline{h} \)>\( {\overline{h}}_{\mathrm{extr}} \)∼\( \frac{c-1}{24} \), of a universal form determined by modular invariance. By including the contribution of light primary operators and multi-twist composites constructed from them in the modular bootstrap, we find that \( {\overline{h}}_{\mathrm{extr}} \) receives corrections in a large spin expansion, which we compute at finite c. The analysis uses a formulation of the modular S-transform as a Fourier transform acting on the density of primary states. For theories with gravitational duals, \( {\overline{h}}_{\mathrm{extr}} \) is interpreted as the extremality bound of rotating BTZ black holes, receiving quantum corrections which we compute at one loop by prohibiting naked singularities in the quantum-corrected geometry. This gravity result is reproduced by modular bootstrap in a semiclassical c → ∞ limit.
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References
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].
D. Poland, S. Rychkov and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques and Applications, Rev. Mod. Phys.91 (2019) 015002 [arXiv:1805.04405] [INSPIRE].
S. Hellerman, A Universal Inequality for CFT and Quantum Gravity, JHEP08 (2011) 130 [arXiv:0902.2790] [INSPIRE].
D. Friedan and C.A. Keller, Constraints on 2d CFT partition functions, JHEP10 (2013) 180 [arXiv:1307.6562] [INSPIRE].
S. Collier, Y.-H. Lin and X. Yin, Modular Bootstrap Revisited, JHEP09 (2018) 061 [arXiv:1608.06241] [INSPIRE].
T. Hartman, D. Mazáč and L. Rastelli, Sphere Packing and Quantum Gravity, arXiv:1905.01319 [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Liouville field theory on a pseudosphere, hep-th/0101152 [INSPIRE].
C.A. Keller and A. Maloney, Poincaré Series, 3D Gravity and CFT Spectroscopy, JHEP02 (2015) 080 [arXiv:1407.6008] [INSPIRE].
J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys.B 270 (1986) 186 [INSPIRE].
N. Afkhami-Jeddi, K. Colville, T. Hartman, A. Maloney and E. Perlmutter, Constraints on higher spin CFT 2, JHEP05 (2018) 092 [arXiv:1707.07717] [INSPIRE].
S. Collier, Y. Gobeil, H. Maxfield and E. Perlmutter, Quantum Regge Trajectories and the Virasoro Analytic Bootstrap, JHEP05 (2019) 212 [arXiv:1811.05710] [INSPIRE].
Y. Kusuki, Light Cone Bootstrap in General 2D CFTs and Entanglement from Light Cone Singularity, JHEP01 (2019) 025 [arXiv:1810.01335] [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].
R.S. Strichartz, A guide to distribution theory and Fourier transforms, World Scientific Publishing Company (2003).
B. Mukhametzhanov and A. Zhiboedov, Modular invariance, tauberian theorems and microcanonical entropy, JHEP10 (2019) 261 [arXiv:1904.06359] [INSPIRE].
S. Carlip, Logarithmic corrections to black hole entropy from the Cardy formula, Class. Quant. Grav.17 (2000) 4175 [gr-qc/0005017] [INSPIRE].
A. Sen, Logarithmic Corrections to Schwarzschild and Other Non-extremal Black Hole Entropy in Different Dimensions, JHEP04 (2013) 156 [arXiv:1205.0971] [INSPIRE].
S. Collier, P. Kravchuk, Y.-H. Lin and X. Yin, Bootstrapping the Spectral Function: On the Uniqueness of Liouville and the Universality of BTZ, JHEP09 (2018) 150 [arXiv:1702.00423] [INSPIRE].
T. Hartman, C.A. Keller and B. Stoica, Universal Spectrum of 2d Conformal Field Theory in the Large c Limit, JHEP09 (2014) 118 [arXiv:1405.5137] [INSPIRE].
B. Ponsot and J. Teschner, Liouville bootstrap via harmonic analysis on a noncompact quantum group, hep-th/9911110 [INSPIRE].
J. Teschner and G. Vartanov, 6j symbols for the modular double, quantum hyperbolic geometry and supersymmetric gauge theories, Lett. Math. Phys.104 (2014) 527 [arXiv:1202.4698] [INSPIRE].
S. Caron-Huot, Analyticity in Spin in Conformal Theories, JHEP09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
D. Simmons-Duffin, D. Stanford and E. Witten, A spacetime derivation of the Lorentzian OPE inversion formula, JHEP07 (2018) 085 [arXiv:1711.03816] [INSPIRE].
P. Kravchuk and D. Simmons-Duffin, Light-ray operators in conformal field theory, JHEP11 (2018) 102 [arXiv:1805.00098] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The Analytic Bootstrap and AdS Superhorizon Locality, JHEP12 (2013) 004 [arXiv:1212.3616] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Universality of Long-Distance AdS Physics from the CFT Bootstrap, JHEP08 (2014) 145 [arXiv:1403.6829] [INSPIRE].
Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, JHEP11 (2013) 140 [arXiv:1212.4103] [INSPIRE].
A.L. Fitzpatrick, E. Katz, D. Poland and D. Simmons-Duffin, Effective Conformal Theory and the Flat-Space Limit of AdS, JHEP07 (2011) 023 [arXiv:1007.2412] [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].
J. Cotler and K. Jensen, A theory of reparameterizations for AdS 3gravity, JHEP02 (2019) 079 [arXiv:1808.03263] [INSPIRE].
A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three Dimensions, JHEP02 (2010) 029 [arXiv:0712.0155] [INSPIRE].
S. Giombi, A. Maloney and X. Yin, One-loop Partition Functions of 3D Gravity, JHEP08 (2008) 007 [arXiv:0804.1773] [INSPIRE].
A.R. Steif, The Quantum stress tensor in the three-dimensional black hole, Phys. Rev.D 49 (1994) 585 [gr-qc/9308032] [INSPIRE].
M. Casals, A. Fabbri, C. Martínez and J. Zanelli, Quantum Backreaction on Three-Dimensional Black Holes and Naked Singularities, Phys. Rev. Lett.118 (2017) 131102 [arXiv:1608.05366] [INSPIRE].
M. Casals, A. Fabbri, C. Martínez and J. Zanelli, Quantum-corrected rotating black holes and naked singularities in (2 + 1) dimensions, Phys. Rev.D 99 (2019) 104023 [arXiv:1902.01583] [INSPIRE].
H. Maxfield, Entanglement entropy in three dimensional gravity, JHEP04 (2015) 031 [arXiv:1412.0687] [INSPIRE].
R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev.D 48 (1993) R3427 [gr-qc/9307038] [INSPIRE].
R.M. Wald and A. Zoupas, A General definition of ‘conserved quantities’ in general relativity and other theories of gravity, Phys. Rev.D 61 (2000) 084027 [gr-qc/9911095] [INSPIRE].
V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev.D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
S. Hollands, A. Ishibashi and D. Marolf, Comparison between various notions of conserved charges in asymptotically AdS-spacetimes, Class. Quant. Grav.22 (2005) 2881 [hep-th/0503045] [INSPIRE].
H. Bremermann, Distributions, complex variables, and Fourier transforms, Addison-Wesley (1965).
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ArXiv ePrint: 1906.04416
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Maxfield, H. Quantum corrections to the BTZ black hole extremality bound from the conformal bootstrap. J. High Energ. Phys. 2019, 3 (2019). https://doi.org/10.1007/JHEP12(2019)003
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DOI: https://doi.org/10.1007/JHEP12(2019)003