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

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) over bar > (h) over bar (extr) similar to 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 (h) over bar (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) over bar (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 -> infinity limit.