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

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Summary

Summary of results

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

The large c partition function
Free energy
Spectrum
Subleading saddles and the enigmatic range
Operator bounds
Angular potential
High and low temperature partition function
Mixed temperature regime
Comparison to 3d gravity
Canonical ensemble
BTZ black holes in the microcanonical ensemble
Enigmatic phases in the microcanonical ensemble
Example: symmetric orbifolds
Partition function
A Density of states in the microcanonical ensemble
B Mixed temperature calculations
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