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

TL;DR

This work analyzes how modular invariance constrains UV data in 2d CFTs with large central charge and sparse light spectra, establishing a universal leading free energy and a Cardy-like high-energy spectrum that extend beyond the conventional Cardy regime. The authors develop a large-$c$ framework, including an angular-potential extension, and derive sharp entropy bounds and universality regions in the $(\beta_L,\beta_R)$ plane, supported by a detailed comparison to 3d gravity (BTZ black holes and Hawking-Page transitions) and by exact results for symmetric orbifolds. They identify an enigmatic range of intermediate energies with nonuniversal entropy but universal upper bounds, mapping these features to nontrivial bulk saddles in holography. The results provide concrete necessary-and-sufficient criteria for holographic behavior in 2d CFTs through leading spectrum and thermodynamics and illustrate them with symmetric orbifolds as maximally dense yet consistent examples.

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, $Δ\to \infty$. 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 $Δ\geq c/6$. 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.

Figures (2)

• Figure 1: Universality in CFT with large $c$ and a sparse light spectrum. (a) Canonical Ensemble: The dashed line ($\beta_L\beta_R = 4\pi^2$) separates high temperatures from low temperatures; in gravity, this would be the Hawking-Page phase transition. We show that the leading free energy is universal and equal to the Cardy value outside of the shaded sliver, and conjecture that this also holds in the sliver. (b) Spectrum: The density of light states in the hatched region is bounded above by the sparseness assumption. We show that the density of states obeys the Cardy formula above the solid curve, and conjecture that this is true above the dashed curve ($E_LE_R = (c/24)^2$). In the enigma range, the entropy is not universal, but satisfies an upper bound that prevents the enigma states from dominating the canonical ensemble.
• Figure 2: Derivation of universal free energy at finite angular potential. We apply an iterative procedure to derive the universal free energy in larger and larger portions of the phase diagram. The shaded regions show the universal regions derived from the first three iterations. After three iterations the universal range encompasses all $(\beta_L,\beta_R)$ away from the white sliver.