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

## TL;DR

This paper calculates quantum corrections to the extremality bound of BTZ black holes using conformal bootstrap techniques in 2D CFTs, linking modular invariance with black hole physics.

## Contribution

It introduces a method to compute quantum corrections to the BTZ extremality bound via modular bootstrap and primary operator contributions in 2D CFTs.

## Key findings

- Quantum corrections to the extremality bound are computed at finite central charge.
- The bootstrap results match gravity calculations in the semiclassical limit.
- Light primary operators influence the density of states at large spin.

## 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 $\bar{h}>\bar{h}_\text{extr}\sim \frac{c-1}{24}$, 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 $\bar{h}_\text{extr}$ 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, $\bar{h}_\text{extr}$ 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 $c\to\infty$ limit.