Universal Spectrum of 2d Conformal Field Theory in the Large c Limit

TL;DR

This paper demonstrates that 2d conformal field theories with large central charge and sparse light spectra exhibit universal thermodynamic and spectral properties across all temperatures, aligning with holographic principles.

Contribution

It establishes that such theories have a universal free energy and spectrum at all temperatures, extending the known high-temperature universality to all regimes, with implications for holography.

Findings

Universal free energy at all temperatures for large c, sparse spectrum theories

Microscopic spectrum matches Cardy entropy for all b4 a0 c/6

Provides criteria for holographic behavior in 2d CFTs

Abstract

Two-dimensional conformal field theories exhibit a universal free energy in the high temperature limit $T \to \infty$, and a universal spectrum in the Cardy regime, $\Delta \to \infty$. We show that a much stronger form of universality holds in theories with a large central charge $c$ and a sparse light spectrum. In these theories, the free energy is universal at all values of the temperature, and the microscopic spectrum matches the Cardy entropy for all $\Delta \geq c/6$. The same is true of three-dimensional quantum gravity; therefore our results provide simple necessary and sufficient criteria for 2d CFTs to behave holographically in terms of the leading spectrum and thermodynamics. We also discuss several applications to CFT and gravity, including operator dimension bounds derived from the modular bootstrap, universality in symmetric orbifolds, and the role of non-universal `enigma' saddlepoints in the thermodynamics of 3d gravity.