Quantum corrections to the BTZ black hole extremality bound from the conformal bootstrap

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 forh >h extr ∼ 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 thath 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,h 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.

Tauberian-Cardy formula with spin

“…The square root edge in h dependence of the density of states should produce a factor of (h − (c − 1)/24) 3/2 . In particular, we choose β = The above result is consistent with the leading result as reported in [4,5,7,8].…”

“…) is a very small number compared to 1 c−1 , this matches with the leading result appeared in the lightcone bootstrap program [4][5][6][7][8] i.e.…”

“…We will see that the finer details of the Cardy formula actually depend on how infinity is approached unless one makes extra assumption about the spectrum. Furthermore, there have been interesting developments in the direction of lightcone bootstrap in recent times [4][5][6][7][8], our analysis puts some of these results on rigorous footing. Another amazing feature is the ability to investigate the "areal" notion of spectral gap.…”

“…In the present case, using the kernel ρ ft-Vir * Now by appropriately choosing β we can recover the "square-root" edge present in the analysis in [7]. The square root edge in h dependence of the density of states should produce a factor of (h − (c − 1)/24) 3/2 .…”

“…While for the upper bound this does not cause any trouble, for the lower bound, it makes thing tricky. In particular, the lower bound on the density of states, appropriately integrated, contains an exponential piece as expected from extended Cardy formula [4,5,7,8] but it comes with a multiplicative order one number, which can become negative unless proven otherwise.…”