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

Henry Maxfield

TL;DR

The paper develops a modular bootstrap framework in 2D CFTs to quantify how light operators and multi-twist composites shift the high-spin extremality bound, connecting this to the BTZ black hole extremality in AdS$_3$. By recasting the modular S-transform as a Fourier transform on the density of primaries and analyzing the spectrum of multi-twist operators, it derives a universal near-extremal density with a spin-dependent edge, and expresses the shift in the extremality bound $ar{h}_ ext{extr}$ in terms of the light operator data. In holographic theories, the authors compute one-loop quantum corrections to the BTZ extremality bound from a bulk scalar and reproduce the same shift via semiclassical modular bootstrap, confirming a consistent CFT–gravity correspondence for near-extremal states. The results generalize to multiple light operators and fermions, and the semiclassical bootstrap demonstrates that the gravity results are encoded in the large-$c$ CFT via MFT/Bose-gas spectral data, highlighting a robust link between modular invariance, high-spin spectra, and quantum gravity in AdS$_3$.

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.

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

TL;DR

The paper develops a modular bootstrap framework in 2D CFTs to quantify how light operators and multi-twist composites shift the high-spin extremality bound, connecting this to the BTZ black hole extremality in AdS. By recasting the modular S-transform as a Fourier transform on the density of primaries and analyzing the spectrum of multi-twist operators, it derives a universal near-extremal density with a spin-dependent edge, and expresses the shift in the extremality bound in terms of the light operator data. In holographic theories, the authors compute one-loop quantum corrections to the BTZ extremality bound from a bulk scalar and reproduce the same shift via semiclassical modular bootstrap, confirming a consistent CFT–gravity correspondence for near-extremal states. The results generalize to multiple light operators and fermions, and the semiclassical bootstrap demonstrates that the gravity results are encoded in the large- CFT via MFT/Bose-gas spectral data, highlighting a robust link between modular invariance, high-spin spectra, and quantum gravity in AdS.

Abstract

Any unitary compact two-dimensional CFT with and no symmetries beyond Virasoro has a parametrically large density of primary states at large spin for , 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 receives corrections in a large spin expansion, which we compute at finite . 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, 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 limit.

Paper Structure

This paper contains 40 sections, 132 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Universal results for unitary compact CFTs
  3. Theories with AdS$_3$ gravitational duals
  4. Organisation of the paper
  5. Acknowledgements
  6. The modular $\mathbb{S}$-transform
  7. The modular S-matrix
  8. Mathematics of the S-transform
  9. Cardy formulas
  10. The modular transform of large spin growth
  11. The spectrum of multi-twist operators
  12. Double-twist Regge trajectories from the fusion kernel
  13. Extending to multi-twists
  14. Multi-twists as multi-particle states in AdS
  15. Corrections to the multi-twist spectrum
  16. ...and 25 more sections

Figures (1)

  • Figure 1: A conjectural cartoon of the spectrum of primaries of an irrational CFT. This paper addresses two features of the spectrum, namely multi-twist operators and the extremality bound, and their relation through modular invariance. The points represent composite multi-twist operators, which grow polynomially in number at large spin. The region above the red dashed line -- the extremality bound -- represents the 'continuum' of operators, with entropy growing as $\sqrt{c\ell}$ at large spin $\ell$ or central charge $c$. The twist of light operators determines the shape of the extremality bound. Generically, we can only trust this picture in an asymptotic expansion in spin, so the region ? without perturbative control is large. In a theory with weakly coupled AdS$_3$ dual, the perturbation theory is instead controlled by large central charge, the validity is extended, and ? is small compared to $c$.