Conditions for Cardy behavior down to the BTZ threshold in large‑c CFTs

Identify precise additional sparsity or spectral conditions, beyond the Hartman–Keller–Stoica bound, that are sufficient to extend the validity of the Cardy formula for the density of states down to the BTZ threshold Δ>c/12 in large‑central‑charge two‑dimensional conformal field theories.

Background

HKS showed that under a sparseness condition, Cardy behavior holds for Δ>c/6 at large c. Gravity predicts Cardy entropy should already apply at the BTZ threshold Δ>c/12.

Bridging this gap requires identifying spectral conditions stronger than HKS, but the exact conditions that guarantee Cardy behavior down to Δ>c/12 are not known, leaving a key piece of the holographic dictionary incomplete.

References

To achieve the extension of the validity region to Δ > c/12, conditions stronger than the HKS bound are required, but what exact conditions lead to Δ > c/12 is still unknown.

— Modern Approach to 2D Conformal Field Theory  (2412.18307 - Kusuki, 2024) in Section 7.3 (Hartman‑Keller‑Stoica Bound and Black Hole Entropy)

Determining how the near-gap excess gives way to BTZ matching at higher energies remains an open problem.

— Large gaps and BTZ entropy in modular spectra with positive integer degeneracies  (2609.39930 - Chang et al., 30 Sep 2026) in Section 1, Introduction; Section 3.3, discussion of the positive-κ family