Quantum corrections to the BTZ black hole extremality bound from the conformal bootstrap
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Springer
Received: August 30, 2019 Accepted: October 2, 2019 Published: December 2, 2019
Henry Maxfield Department of Physics, University of California, Santa Barbara, CA 93106, U.S.A.
E-mail: [email protected] 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 ¯>h ¯ extr ∼ c−1 , of a universal form determined by modular invariance. By including the h 24 contribution of light primary operators and multi-twist composites constructed from them ¯ extr receives corrections in a large spin expansion, in the modular bootstrap, we find that h 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 ¯ extr is interpreted as the extremality bound of rotating BTZ black holes, receiving duals, h 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. Keywords: AdS-CFT Correspondence, Black Holes, Conformal and W Symmetry, Conformal Field Theory ArXiv ePrint: 1906.04416
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP12(2019)003
JHEP12(2019)003
Quantum corrections to the BTZ black hole extremality bound from the conformal bootstrap
Contents 1 2 4 5
2 The 2.1 2.2 2.3 2.4
modular S-transform The modular S-matrix Mathematics of the S-transform Cardy formulas The modular transform of large spin growth
5 5 7 8 9
3 The 3.1 3.2 3.3 3.4
spectrum of multi-twist operators Double-twist Regge trajectories from the fusion kernel Extending to multi-twists Multi-twists as multi-particle states in AdS Corrections to the multi-twist spectrum
10 10 12 14 14
4 The 4.1 4.2 4.3 4.4
S-transform of multi-twist operators Shift in extremality bound from universal multi-twists Multiple operators Fermions Anomalous dimensions
15 16 18 19 19
5 Semiclassical AdS3 gravity 5.1 BTZ and the extremality bound 5.2 Including a scalar field at one loop 5.3 Conserved quantities 5.4 The modified extremality bound
21 21 22 24 25
6 Semiclassical bootstrap
27
A The A.1 A.2 A.3 A.4 A.5
28 28 30 30 32 33
mathematical appendix Distributions A Fourier transform Analytic representations A distributional binomial theorem Asymptotics of distributions
–i–
JHEP12(2019)003
1 Introduction 1.1 Universal results for unitary compact CFTs 1.2 Theories with AdS3 gravitational duals 1.3 Organisation of the paper
B Counting multi-trace states B.1 Bosons B.2 Counting at large spin B.3 Multiple species B.4 Fermions
35 35 36 37 37
C The covariant phase space formalism C.1 The formalism C.2 Einstein-Hilbert
38 38 39
Introduction
The bootstrap program applied to two-dimensional conformal field theory has been remarkably successful for the classification and solution of rational theories [1], but far less progress has been made
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