Universal Spectrum of 2d Conformal Field Theory in the Large c Limit (1405.5137v2)

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.

• The paper establishes that large-c 2D CFTs with sparse spectra exhibit a universal free energy valid across all temperature scales.
• The paper demonstrates that the Cardy formula applies well beyond high-energy limits by unifying the asymptotic density of states through modular invariance.
• The paper links bounds on medium-energy states in these theories to holographic insights from 3D gravity, enhancing our understanding of black hole microstates.

Universal Spectrum of 2D Conformal Field Theory

The paper "Universal Spectrum of 2d Conformal Field Theory" by Thomas Hartman, Christoph A. Keller, and Bogdan Stoica presents a detailed investigation into the universality features of two-dimensional conformal field theories (2D CFTs) with large central charge and sparse low-lying spectra. The authors explore conditions under which these CFTs exhibit a universal free energy, aligning with known gravitational physics in AdS$_3$ contexts and providing insights into UV/IR connections.

Summary

The paper focuses on understanding under what circumstances a 2D CFT can be described by a universal spectrum, particularly demonstrating that for theories with large central charge ($c$) and sparse light spectra, the Cardy formula for entropy holds beyond its traditional region and describes the microscopic spectrum for energies $\Delta \geq \frac{c}{6}$. The same universality conditions apply to three-dimensional quantum gravity, suggesting that certain $2d$ CFTs behave holographically by encapsulating gravitational insights through modular invariance and spectrum considerations.

Key Findings

1. Universality of Free Energy: The authors establish that for a class of 2D CFTs, a universal form of the free energy can be derived at all temperatures. This result was traditionally understood in the high-temperature limit, but the paper extends it universally across temperature scales.
2. Cardy Formula Universality: The Cardy formula, which gives asymptotic density of states, is shown to hold beyond the usual high-energy, large-$c$ regime. For these theories, the conditions are met uniformly for almost all energy scales, unified under the modular invariance of the partition function.
3. Spectrum Bounds: Introducing a classification of states as light, medium, or heavy based on their energy, the authors highlight bounds on the density of states within the "enigmatic" range where energies fall between $0 < E < c/12$. Here too, the universal features of $3d$ gravity align with their findings, suggesting that medium energy states remain thermodynamically unstable analogous to certain gravitational configurations.
4. Symmetric Orbifolds: Applying their results to symmetric orbifolds, the paper verifies that these theories universally exhibit the derived free energy and spectral distributions. This matches historical insights into black hole microstate counts in string theoretic models, emphasizing their universal nature.

Implications and Future Work

The implications are manifold, encompassing both theoretical enrichment and practical aspects of model-building in string theory and holography. By tying together modular invariance with gravitational universality, the authors provide a framework for understanding how classical gravitational features might emerge from fundamentally quantum configurations in 2D conformal theories.

Further research can extend these results to other forms of CFTs, probing the boundaries of the derived universality and exploring how these findings might generalize to higher-genus surfaces or alternative holographic dual configurations. The findings also motivate a deeper look into non-holomorphic theories and the roles played by enigmatic states in the context of complex dynamical systems.

In conclusion, the paper contributes significantly to theoretical physics by advancing the understanding of universality in two-dimensional conformal field theories, revealing intricate connections with three-dimensional gravity and holography, and establishing ground for further explorations into the nature of space-time and fundamental forces.