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.
Highlights
We show that a much stronger form of universality holds in theories with a large central charge c and a sparse light spectrum
The free energy is universal at all values of the temperature, and the microscopic spectrum matches the Cardy entropy for all
The same is true of three-dimensional quantum gravity; our results provide simple necessary and sufficient criteria for 2d CFTs to behave holographically in terms of the leading spectrum and thermodynamics
Summary
Operators in a unitary 2d CFT are labeled by their left and right conformal weights (h, h) with h, h ≥ 0. The leading order spectrum of 3d gravity plus matter (or gravity on AdS3 × X) in this range is non-universal, because in addition to the usual BTZ black holes there can be entropically dominant ‘enigmatic’ black holes [18, 19]. These solutions, discussed, obey the bound (1.13). Density of states is completely universal for all symmetric orbifold theories, and saturates the bounds (1.10), (1.13) and (1.15) In this sense, symmetric orbifolds have the maximally dense spectrum compatible with 3d gravity
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